<PAGE>   1
                                  UNITED STATES
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                                    FORM 10-Q

(Mark One)

[X]      QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES
         EXCHANGE ACT OF 1934 For the period ended June 30, 1998

                                       or

[ ]      TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES
         EXCHANGE ACT OF 1934 
         For the Transition period from _________________ to ________________ 
         Commission File Number 33-16973
                                --------

                                   NET 1 L. P.
             ------------------------------------------------------
             (Exact name of Registrant as specified in its charter)

                  Delaware                                    13-3421566
       -------------------------------                    -------------------
       (State or other jurisdiction of                     (I.R.S. Employer
       incorporation or organization)                     Identification No.)

  c/o Lexington Corporate Properties Trust
            355 Lexington Avenue
                New York,  NY                                     10017
  ---------------------------------------                       ----------
  (Address of principal executive offices)                      (Zip Code)

Registrant's telephone number, including area code           (212) 692-7200
                                                             --------------

Securities registered pursuant to Section 12(b) of the Act:   None

Securities registered pursuant to Section 12(g) of the Act: Units of Limited
Partnership Interests

Indicate by check mark whether the Registrant (1) has filed all reports required
to be filed by Section 13 or 15(d) of the Securities Exchange Act of 1934 during
the preceding 12 months (or for such shorter period that the Registrant was
required to file such reports), and (2) has been subject to such filing
requirements for the past 90 days.
                                  Yes x .  No___.

State the aggregate market value of the voting stock held by non-affiliates of
the Registrant. Not applicable.

There is no active public market for the units of limited partnership interests
issued by the Registrant.



<PAGE>   2
                       PART 1. - FINANCIAL INFORMATION
                       -------------------------------
                                      
                         ITEM 1. FINANCIAL STATEMENTS
                         ----------------------------

                  NET 1 L. P. AND CONSOLIDATED PARTNERSHIPS
                                      
                         CONSOLIDATED BALANCE SHEETS
                                    ($000)
                                      
               June 30, 1998 (Unaudited) and December 31, 1997



                                     ASSETS
                                     ------
<TABLE>
<CAPTION>
                                                                June 30,         December 31,
                                                                  1998               1997
                                                                --------         ------------
<S>                                                             <C>                <C>
Real estate, at cost                                            $ 26,440           $ 26,440
Less:  accumulated depreciation                                    2,972              2,735
                                                                --------           --------
                                                                  23,468             23,705

Cash and cash equivalents                                          1,441              1,312
Rent receivable                                                      442                400
Other assets                                                         410                246
                                                                --------           --------

                                                                $ 25,761           $ 25,663
                                                                ========           ========



                      LIABILITIES AND PARTNERS' CAPITAL
                      ---------------------------------

Mortgage notes payable                                          $  5,514           $  5,676
Accrued interest payable                                              29                 30
Accounts payable and other liabilities                                63                165
                                                                --------           --------
                                                                   5,606              5,871
                                                                --------           --------

Partners' capital (deficit):
   General Partner                                                  (167)              (174)
   Limited Partners ($1,000 per Unit,
   50,000 Units authorized, 30,772
   Units issued and outstanding)                                  20,322             19,966
                                                                --------           --------
   Total partners' capital                                        20,155             19,792
                                                                --------           --------

                                                                $ 25,761           $ 25,663
                                                                ========           ========
</TABLE>

     See accompanying notes to unaudited consolidated financial statements.

<PAGE>   3
                    NET 1 L. P. AND CONSOLIDATED PARTNERSHIPS

                        CONSOLIDATED STATEMENTS OF INCOME
                         ($000 EXCEPT PER UNIT AMOUNTS)

                    Quarters Ended June 30, 1998 and 1997 and
                     Six Months Ended June 30, 1998 and 1997
                                   (Unaudited)



<TABLE>
<CAPTION>
                                                                                    Six Months           Six Months
                                         Quarter Ended       Quarter Ended             Ended                Ended
                                            June 30,            June 30,              June 30,             June 30,
                                              1998                1997                  1998                 1997
                                             ------               ------               ------               ------

Revenues:
<S>                                          <C>                  <C>                  <C>                  <C>   
   Rental                                    $  703               $  649               $1,789               $1,549
   Interest and other                            21                   25                   39                   53
                                             ------               ------               ------               ------

                                                724                  674                1,828                1,602
                                             ------               ------               ------               ------

Expenses:

   Interest expense                             128                  108                  259                  202
   Depreciation                                 119                  105                  237                  203
   General, administrative
   and other                                     95                   96                  183                  195
                                             ------               ------               ------               ------

                                                342                  309                  679                  600
                                             ------               ------               ------               ------

Net income                                   $  382               $  365               $1,149               $1,002
                                             ======               ======               ======               ======


Net income per Unit of limited
   partnership interest                      $12.16               $11.63               $36.59               $31.92
                                             ======               ======               ======               ======
</TABLE>


     See accompanying notes to unaudited consolidated financial statements.
<PAGE>   4
                    NET 1 L. P. AND CONSOLIDATED PARTNERSHIPS

                      CONSOLIDATED STATEMENTS OF CASH FLOWS
                                     ($000)

                     Six months ended June 30, 1998 and 1997
                                   (Unaudited)




<TABLE>
<CAPTION>
                                                                Six Months              Six Months
                                                                   Ended                   Ended
                                                                June 30, 1998           June 30, 1997
                                                                -------------           -------------
<S>                                                                <C>                    <C>    
Cash flows from operating activities:
   Net income                                                      $ 1,149                $ 1,002
                                                                   -------                -------
   Adjustments to reconcile net income to
       net cash provided by operating activities:
       Depreciation                                                    237                    203
       Other, net                                                     (309)                   (72)
                                                                   -------                -------
       Total adjustments                                               (72)                   131
                                                                   -------                -------

   Net cash provided by operating activities                         1,077                  1,133
                                                                   -------                -------

Cash flows from investing activities:
   Acquisition of Texas Property, net of
       mortgage liability assumed                                     --                   (1,204)
                                                                   -------                -------
   Net cash used in investing activities                              --                   (1,204)
                                                                   -------                -------

Cash flows from financing activities:
   Principal payments on mortgage notes                               (162)                   (86)
   Cash distributions to partners                                     (786)                  (786)
                                                                   -------                -------
   Net cash used in financing activities                              (948)                  (872)
                                                                   -------                -------

Net increase (decrease) in cash and cash equivalents                   129                   (943)

Cash and cash equivalents at beginning of period                     1,312                  2,123
                                                                   -------                -------

Cash and cash equivalents at end of period                         $ 1,441                $ 1,180
                                                                   =======                =======

Supplemental disclosure of cash flow information:

Cash payments for interest                                         $   260                $   189
                                                                   =======                =======
</TABLE>


     See accompanying notes to unaudited consolidated financial statements.


<PAGE>   5
                    NET 1 L. P. AND CONSOLIDATED PARTNERSHIPS

                   NOTES TO CONSOLIDATED FINANCIAL STATEMENTS

                                  June 30, 1998
                                   (Unaudited)



1.       The Partnership and Basis of Presentation
         -----------------------------------------

         Net 1 L. P. (the "Partnership") was formed as a limited partnership on
         August 25, 1987 under the laws of the State of Delaware to invest in
         real estate or interests therein to be net leased to corporations or
         other entities.

         As of June 30, 1998, the Partnership has a total of 30,772 Units issued
         and outstanding held by approximately 1,500 limited partners.

         The unaudited financial statements reflect all adjustments that are, in
         the opinion of the General Partner, necessary to a fair statement of
         the results for the interim period presented. For a more complete
         understanding of the Partnership's financial position and accounting
         policies, reference is made to the financial statements previously
         filed with the Securities and Exchange Commission with the
         Partnership's Annual Report on Form 10-K for the year ended December
         31, 1997.

         Management of the Partnership has made a number of estimates and
         assumptions relating to the reporting of assets and liabilities and the
         disclosure of contingent assets and liabilities to prepare these
         financial statements in conformity with generally accepted accounting
         principles. Actual results could differ from those estimates.

2.       Summary of Significant Accounting Policies
         ------------------------------------------

         Net income per Unit amounts were calculated by using the weighted
         average number of Units outstanding for each period and allocating the
         income attributable for that period to the Limited Partners. The
         weighted average number of Units outstanding was 30,772 for all periods
         presented.

3.       The Partnership Agreement
         -------------------------

         For financial statement reporting purposes all items of income are
         allocated in the same proportion as distributions of distributable
         cash.

         Distributable cash attributed to a particular limited partner's Unit is
         calculated from the date of admission to the Partnership. The unpaid
         cumulative preferred return at June 30, 1998 totaled $11.779 million
         ($379.55 to $385.40 per Unit, per close). On July 31, 1998, the unpaid
         cumulative preferred return at June 30, 1998 was reduced by a cash
         distribution to the Limited Partners for the quarter ended June 30,
         1998 totaling $384,958 ($12.51 per Unit). The General Partner received
         a cash distribution of $7,856 on July 31, 1998.



<PAGE>   6
                    NET 1 L. P. AND CONSOLIDATED PARTNERSHIPS

                   NOTES TO CONSOLIDATED FINANCIAL STATEMENTS



4.       Related Party Transactions
         --------------------------

         Leased Properties Management, Inc., an affiliate of the General
         Partner, is entitled to receive a fee for managing the Partnership's
         properties in the amount of 1% of gross annual rental receipts (or a
         greater amount in certain circumstances). For the six months ended June
         30, 1998 and 1997, property management fees of $17,000 and $15,000,
         respectively had been incurred.






<PAGE>   7
                 ITEM 2. MANAGEMENT'S DISCUSSION AND ANALYSIS
               OF FINANCIAL CONDITION AND RESULTS OF OPERATIONS
               ------------------------------------------------


Liquidity and Capital Resources
-------------------------------

The Partnership attempts to maintain a working capital reserve in an amount
equal to 3% of the gross proceeds of its offering, an amount which is
anticipated to be sufficient to satisfy liquidity requirements. Liquidity of the
Partnership could be adversely affected by unanticipated costs, lessees
experiencing financial difficulties and greater than anticipated operating
expenses. To the extent that such working capital reserves are insufficient to
satisfy the cost requirements of the Partnership, additional funds may be
obtained through short-term or permanent loans or by reducing distributions to
limited partners.

The unpaid cumulative preferred return at June 30, 1998 totaled $11.779 million
($379.55 to $385.40 per Unit, per close), and was reduced by $384,958 ($12.51
per Unit) with the second quarter 1998 distribution paid in July 1998.

Except for the debt service requirements under the mortgages, there are no
material restrictions upon the Partnership's present or future ability to make
distributions in accordance with the provisions of its Partnership Agreement.

Impact of Year 2000
-------------------

The Partnership is evaluating its computer and communication systems that could
be affected by the "Year 2000" issue. The year 2000 problem is the result of
computer programs being written using two digits rather than four to define the
applicable year. Any of the Partnership's programs that have time-sensitive
software may recognize a date using "00" as the year 1900 rather than the year
2000. This could result in a major system failure or miscalculation. The
Partnership presently believes that the year 2000 problem will not pose
operational problems for the Partnership's computer and communication systems
and will not have a material impact on the operations of the Partnership.



<PAGE>   8


Results of Operations ($000)
----------------------------
<TABLE>
<CAPTION>
                                                                                     Increase (Decrease)
                                    Quarters                Six months           Quarters       Six months
                                  ended June 30,           ended June 30,      ended June 30,  ended June 30,
                                1998         1997         1998         1997         1998          1998
                                ----         ----         ----         ----         ----          ----
<S>                            <C>          <C>          <C>          <C>          <C>           <C>   
Total revenues                 $  724       $  674       $1,828       $1,602       $   50        $  226
                               ------       ------       ------       ------       ------        ------

Total expenses
 Interest                         128          108          259          202           20            57
 Depreciation                     119          105          237          203           14            34
 General, administrative
  and other                        95           96          183          195           (1)          (12)
                               ------       ------       ------       ------       ------        ------
                                  342          309          679          600           33            79
                               ------       ------       ------       ------       ------        ------

Net income                     $  382       $  365       $1,149       $1,002       $   17        $  147
                               ======       ======       ======       ======       ======        ======
</TABLE>

The changes in results of operations with respect to revenues, interest and
depreciation for the quarter and six months ended June 30, 1998 are primarily
attributable to the operations of real property investment acquired in the
second quarter of 1997.

Accounting Standards Issued but not yet Adopted
-----------------------------------------------

In June, 1998, the Financial Accounting Standards Board issued Statement of
Financial Accounting Standards No. 133 ("SFAS 133") "Accounting for Derivative
Instruments and Hedging Activities". The Statement establishes accounting and
reporting standards for derivative instruments and hedging activities. This
Statement is effective for all fiscal quarters of fiscal years beginning after
June 15, 1999. The adoption of SFAS 133 is not expected to have any impact on
the financial position or results of operations of the Partnership.


               ITEM 3. QUANTITATIVE AND QUALITATIVE DISCLOSURES
                              ABOUT MARKET RISK
               ------------------------------------------------

                               Not applicable.


<PAGE>   9
                         PART II - OTHER INFORMATION
                         ---------------------------

ITEM 1.  Legal Proceedings - not applicable.

ITEM 2.  Changes in Securities - not applicable.

ITEM 3. Defaults under the Senior Securities - not applicable.

ITEM 4. Submission of Matters to a Vote of Security Holders - not applicable.

ITEM 5.  Other Information - not applicable.

ITEM 6.  Exhibits and Reports on Form 8-K.

                  (a)      Exhibits.
                           Exhibit No.                        Exhibit
                           -----------                        -------
                                    27               Financial Data Schedule

                  (b)      Reports on form 8-K filed during the second quarter
                           ended June 30, 1998. None.


<PAGE>   10
                                  SIGNATURES
                                  ----------

Pursuant to the requirements of the Securities Exchange Act of 1934, the
registrant has duly caused this report to be signed on its behalf by the
undersigned thereunto duly authorized.

                                NET 1 L. P.

                                By:   Lepercq Net 1 L. P. 
                                      --------------------
                                      its general partner

                                By:   Lepercq Net 1 Inc.  
                                      --------------------
                                      its general partner


Date: August 13, 1998           By:   /s/E. Robert Roskind
      ---------------                 --------------------
                                      E. Robert Roskind
                                      President