<PAGE>   1
                                                                   Exhibit 10.13

                              EMPLOYMENT AGREEMENT

         EMPLOYMENT AGREEMENT between The Carbide/Graphite Group, Inc., a
Delaware corporation (the "Corporation") and Stephen D. Weaver (the "Executive")
dated as of February 1, 2000 (the "Agreement").

         WHEREAS, the Corporation and the Executive have entered into an
employment agreement dated February 1, 1998 which is superseded by this
Agreement; and

         WHEREAS, the Corporation wishes to employ the Executive as Vice
President-Finance/Chief Financial Officer of the Corporation on the terms set
forth herein and the Executive wishes to be employed by the Corporation on such
terms; IT IS, THEREFORE AGREED:

         1. Employment. The Corporation hereby agrees to employ the Executive,
and the Executive hereby agrees to be employed by the Corporation, for the
period commencing as of the date hereof and ending on January 31, 2002 as its
Vice President-Finance/Chief Financial Officer (the "Employment Period").

         2. Duties. During the Employment Period, the Executive agrees to devote
his attention full time and during normal business hours to his responsibilities
as Vice President-Finance/Chief Financial Officer of the Corporation and to use
his best efforts to perform faithfully and efficiently such responsibilities.
The Executive shall, subject to the supervision and control of the Chairman/CEO
of the Corporation, perform such duties and exercise such supervision and powers
over and with regard to the business of the Corporation as are contemplated to
be performed by the Vice President-Finance/CFO pursuant to the By-laws of the
Corporation and such additional duties as may from time to time be prescribed by
the Chairman/CEO and the Board of Directors.

<PAGE>   2


         3. Base Compensation. During the Employment Period, the Executive shall
receive a base salary at an annual rate of at least $206,000 with any increase
thereto to be determined by the Compensation Committee of the Board of Directors
from time to time.

         4. Stock Options. During the Employment Period, the Executive shall
receive stock option grants as determined by the Stock Option Plan Committee on
terms and conditions consistent with other participants in the Stock Option
Plan.

         5. Annual Incentive Awards. Subject to the terms of the Corporation's
Annual Incentive Bonus Plan, the Executive's participation in the Plan for the
fiscal years ending July 31, 2000 and July 31, 2001 and the period August 1,
2001 to January 31, 2002 shall be at a target award level equal to a rate of 30%
of base pay, a "near-miss" award level of 15% of base pay and a maximum award
level of 60% of base pay.

         6. Termination. (a) Death or Disability. This Agreement shall terminate
automatically upon the Executive's death. The Corporation may terminate this
Agreement during the Employment Period after having established the Executive's
"Disability" (as defined below), by giving the Executive written notice of its
intention to terminate the Executive's employment. For purposes of this
Agreement, "Disability" means the Executive's inability to substantially perform
his duties and responsibilities to the Corporation by reason of a physical or
mental disability or infirmity (i) for a continuous period of six months or (ii)
at such earlier time as the Executive submits medical evidence satisfactory to
the Corporation that the Executive has a physical or mental disability or
infirmity that will likely prevent the Executive from substantially performing
his duties and responsibilities for six months or longer. The date of Disability
shall be the day on which the Executive receives notice from the Corporation
pursuant to this Section 6(a).


                                       2
<PAGE>   3


                  (b) Cause. The Corporation shall have the right to terminate
the Executive's employment for "Cause" during the Employment Period. For
purposes of this Agreement, "Cause" shall mean (i) the willful and continued
failure by the Executive to perform substantially his duties to the Corporation
or its subsidiaries (other than any such failure resulting from his Disability)
within a reasonable period of time after a written demand for substantial
performance is delivered to the Executive by the Chairman/CEO, which demand
specifically identities the manner in which the Chairman/CEO believes that the
Executive has not substantially performed his duties, (ii) embezzlement or theft
from the Corporation or any subsidiary or affiliate by the Executive or the
commission or perpetration by the Executive of any act involving moral
turpitude, or (iii) any material and willful violation by the Executive of his
obligations under Section 8 hereof.

                  (c) Change of Control. The Executive shall have the right to
terminate this Agreement during the Employment Period upon thirty (30) days
prior written notice to the Corporation or a successor of the Corporation, as
the case may be, for "Good Reason" on or subsequent to the consummation of a
"Change of Control". For purposes of this Agreement, (A) "Good Reason" shall
mean (1) a change in the Executive's duties and responsibilities without his
consent such that his duties and responsibilities are materially reduced or
altered in a manner unfavorable to him or a decrease in the Executive's salary
or bonus award potential, or a material decrease in his benefits or (2) a change
in the location at which the Executive's duties are principally carried out of
more than 20 miles from the Corporation's principal executive offices in
Pittsburgh, Pennsylvania or (3) the Corporation or its successor is unwilling to
extend the Executive's employment upon terms at least as favorable to the
Executive as the terms set forth in this agreement; and (B) "Change of Control"
shall mean (i) a change in control of the Board of


                                       3
<PAGE>   4

Directors of the Corporation pursuant to which any single Person or two or more
Persons acting in concert acquires control of such Board of Directors or (ii)
the Transfer of a least 51% or more of the voting equity interests in the
Corporation (or any parent of the Corporation), whether by sale, merger,
consolidation or otherwise, to any single Person or two or more Persons acting
in concert; provided that two or more Persons shall be considered to be acting
in concert for purposes of clauses (i) and (ii) hereof only if such Persons
would have been considered to be acting in concert as a "group" for purposes of
Section 13(d) of the Securities Exchange Act of 1934, for such purposes treating
voting equity interests of the Corporation held or acquired by such Persons as
if such voting equity interests were equity securities in respect of which a
Schedule 13D would be required to be filed with the Securities and Exchange
Commission and as if the requisite percentage and other threshold conditions to
such filing were satisfied. "Person" means any individual, corporation,
partnership, joint venture, association, joint-stock company, trust,
unincorporated organization or governmental body. "Transfer" means, sell,
transfer, convey, lease and/or deliver (other tenses of the term have similar
meaning) or sale, transfer, assignment, conveyance, lease and/or delivery, as
indicated by the context.


         7. Effect of Termination. (a) Upon termination of the Executive's
employment during the Employment Period, because of death or Disability as
provided in subsection 6(a), following termination of the Executive's employment
the Corporation shall continue to pay the Executive or his estate or other
personal representative as severance (x) the amount of the Executive's salary as
provided in Section 3 at the rate in effect immediately prior to termination of
his employment for a period of twenty-four months less the amount of any
disability payments made by the Corporation or any Corporation plan and (y) in
the case of a Disability termination will afford to the Executive at the
Corporation's expense, health insurance


                                       4
<PAGE>   5

benefits (including medical and dental) and life insurance equivalent to the
benefits enjoyed by the Executive at the date of termination (the "Insurance
Benefits') for a period of twenty-four months from the date of such termination.

                  (b) If the Executive's employment is terminated by the
Corporation during the Employment Period (other than for death, Disability or
Cause or other than by virtue of a Change of Control pursuant to subsection
6(c)), or in the event the provisions of subsection 6(c)(A)(3) are not
applicable and the Corporation is unwilling to extend the Executive's employment
at the expiration of this Agreement upon terms at least as favorable to the
Executive as the terms set forth herein, the Corporation shall pay to the
Executive a cash amount as severance in a lump sum equal to twice the
Executive's annual base salary then in effect pursuant to Section 3 plus an
amount equal to twice the average of the previous two years of bonus plan
payouts for the Executive and shall continue to maintain the Insurance Benefits
for a period of 24 months from the date of such termination or expiration. The
lump sum payment shall be paid to the Executive within 45 days after the date of
such termination or expiration.


                  (c) If the Executive's employment is terminated by the
Executive during the Employment Period pursuant to subsection 6(c) (including a
termination arising from the circumstances set forth in subsection 6(c)(A)(3)),
the Corporation shall pay to the Executive a cash amount as severance in a lump
sum equal to 2.99 times the Executive's base salary then in effect pursuant to
Section 3 plus 2.99 times the average of the previous two years of bonus plan
payouts for the Executive; provided that such payment under this subsection 7(c)
shall be limited to an amount which would not constitute an "excess parachute
payment" under Section 280G of the federal Internal Revenue Code. Such lump sum
payment shall be paid within 45 days after the date of termination provided for
in subsection 6(c). The Corporation shall continue to


                                       5
<PAGE>   6


maintain the Insurance Benefits for a period of 24 months from the date of the
termination provided for in subsection 6(c).

                  (d) Nothing herein shall be deemed to restrict or reduce the
Executive's vested benefits under the Corporation's Stock Option Plans, the
compensation deferral plan, the Savings Incentive Plan or the Annual Incentive
Bonus Plan as determined in accordance with the provisions of such plans.

                  (e) No continued salary or severance shall be paid if the
Executive's employment terminates for any reason during the Employment Period
other than as set forth above in this Section 7.

         8. Confidential Information; Non-Competition. (a) For a three year
period commencing on the date on which the Executive's employment with the
Corporation, or any of its affiliates, terminates for whatever reason (the "Date
of Termination"), (i) the Executive shall hold in a fiduciary capacity for the
benefit of the Corporation all secret or confidential information, knowledge or
data relating to the Corporation or its affiliates, and their respective
businesses which shall not be public knowledge (other than information which
becomes public as a result of acts of the Executive or his representatives in
violation of this Agreement), including without limitation, customer lists, bid,
proposals, contracts, matters subject to litigation, technology or financial
information of the Corporation or its subsidiaries and other know-how, and (ii)
the Executive shall not, without the prior written consent of the Corporation,
communicate or divulge any such information, knowledge or data to anyone other
than the Corporation and those designated by it in writing.

                  (b) For a three year period commencing on the Date of
Termination, the Executive will not, directly or indirectly, (i) own, manage,
operate, control or participate in


                                       6
<PAGE>   7

the ownership, management or control of, or be connected as an officer,
employee, partner, director, or consultant or otherwise, with, or have any
financial interest in (except for (A) ownership as of the date hereof, (B) any
ownership in the common stock of the Corporation, or (C) any ownership of less
than 5% of the outstanding equity interest in any entity) (I) the manufacture of
graphite electrodes, or (II) the refining of products required for production
of, or the production of, needle coke, or the manufacture or marketing of
calcium carbide and its direct derivatives, in each case, in any market in which
the Corporation or its affiliates conducts or solicits business or (ii) solicit
or contact any employee of the Corporation or its affiliates with a view to
inducing or encouraging such employee to leave the employ of the Corporation or
its affiliates for the purpose of being employed by the Executive, an employer
affiliated with the Executive, or any competitor of the Corporation or any
affiliate thereof; provided that the provisions of subsection (b)(i) shall be
limited to a period of two years in the event the Corporation is unwilling to
extend the Executive's employment at the expiration of this Agreement upon terms
at least as favorable to the Executive as set forth in this Agreement or the
termination by the Corporation of the Executive's employment during the
Employment Period other than for Cause or Disability.


                  (c) The Executive acknowledges that the provisions of this
Section 8 are reasonable and necessary for the protection of the Corporation and
that the Corporation will be irrevocably damaged if such provisions are not
specifically enforced. Accordingly, the Executive agrees that, in addition to
any other relief to which the Corporation may be entitled in the form of actual
or punitive damages, the Corporation shall be entitled to seek and obtain
injunctive relief from a court of competent jurisdiction (without the posting of
a bond therefor)


                                       7
<PAGE>   8

for the purposes of restraining the Executive from any actual or threatened
breach of such provisions.


         9. Successors. (a) This Agreement is personal to the Executive and
without the prior written consent of the Corporation shall not be assignable by
the Executive other than by will or the laws of descent and distribution. This
Agreement shall inure to the benefit of and be enforceable by the Executive's
legal representatives.

                  (b) This Agreement shall inure to the benefit of and be
binding upon the Corporation and its successors.

                  (c) This Agreement supersedes the employment agreement dated
February 1, 1998 between the Executive and the Corporation which prior agreement
is no longer in force.

         10. Miscellaneous. (a) This Agreement shall be governed by and
construed in accordance with the laws of the State of Delaware , without
reference to principles of conflict of laws.

                  (b) All notices and other communications hereunder shall be in
writing and shall be given by hand delivery to the other party or by registered
or certified mail, return receipt requested, postage prepaid, addressed as
follows:

                  If to the Executive:

                  Mr. Stephen D. Weaver
                  259 Trotwood Drive
                  Pittsburgh, Pennsylvania  15241

                  If to the Corporation:

                  The Carbide/Graphite Group, Inc.
                  One Gateway Center, 19th Floor
                  Pittsburgh, PA  15222

                  Attention:  Secretary


                                       8
<PAGE>   9

or to such other address as either party shall have furnished to the other in
writing in accordance herewith. Notices and communications shall be effective
when actually received by the addressee.

                  (c) The Corporation may withhold from any amounts payable
under this Agreement such Federal, state or local taxes as shall be required to
be withheld pursuant to any applicable law or regulation.

                  (d) The Corporation's failure to insist upon strict compliance
with any provision hereof shall not be deemed to be a waiver of such provision
or any other provision hereof. The Executive's failure to insist upon strict
compliance with any provision hereof shall not be deemed to be a waiver of such
provision hereof.

                  (e) This Agreement embodies the entire agreement between the
parties with respect to the Executive's employment, and may not be changed or
terminated orally.

         IN WITNESS WHEREOF, the Executive has hereunto set his hand and
pursuant to the authorization from its Board of Directors the Corporation has
caused these presents to be executed in its name and on its behalf, all as of
the day and year first above written.

                                        THE CARBIDE/GRAPHITE GROUP, INC.

                                        By    /s/   Walter B. Fowler
                                           -------------------------------
                                        Name:  Walter B. Fowler
                                        Title: CEO


                                              /s/   Stephen D. Weaver
                                             ----------------------------
                                                    Stephen D. Weaver


                                       9
<PAGE>   10






                                                              June 5, 2000


Mr. Stephen D. Weaver
259 Trotwood Drive
Pittsburgh, Pennsylvania  15241

Dear Steve:

                  Reference is made to the Employment Agreement dated as of
February 1, 2000 between The Carbide/Graphite Group, Inc. and yourself (the
"Employment Agreement"), and in particular to paragraphs 1 through 3 thereof. In
light of your promotion to "Senior Vice President- Electrodes and Graphite
Specialty Products", the references in paragraphs 1 and 2 to your title are
deemed amended, effective as of May 1, 2000, to such title which shall be
substituted for the title of "Vice President/Chief Financial Officer".
Furthermore, effective as of May 1, 2000 your base salary as set forth in
paragraph 3 of the Employment Agreement shall be $216,000.

                  On behalf of the Board of Directors of the Company, I would
like to express our pleasure at your promotion.

                                          Best regards,

                                          Sincerely,

                                          /s/ Walter B. Fowler
                                          ------------------------------
                                          Walter B. Fowler
                                          Chairman and Chief Executive Officer

Agreed:

/s/  Stephen D. Weaver
----------------------------
Stephen D. Weaver


                                       10