<Page>
                                                                    EXHIBIT 31.1

                                 CERTIFICATION

I, John C. Malone, certify that:

1.  I have reviewed this quarterly report on Form 10-Q of Discovery Holding
    Company;

2.  Based on my knowledge, this quarterly report does not contain any untrue
    statement of a material fact or omit to state a material fact necessary to
    make the statements made, in light of the circumstances under which such
    statements were made, not misleading with respect to the period covered by
    this quarterly report;

3.  Based on my knowledge, the financial statements and other financial
    information included in this quarterly report fairly present in all material
    respects the financial condition, results of operations and cash flows of
    the registrant as of, and for, the periods presented in this quarterly
    report;

4.  The registrant's other certifying officers and I are responsible for
    establishing and maintaining disclosure controls and procedures (as defined
    in Exchange Act Rules 13a-15(e) and 15d-15(e)) for the registrant and we
    have:

    a)  designed such disclosure controls and procedures, or caused such
       disclosure controls and procedures to be designed under our supervision,
       to ensure that material information relating to the registrant, including
       its consolidated subsidiaries, is made known to us by others within those
       entities, particularly during the period in which this quarterly report
       is being prepared;

    b)  evaluated the effectiveness of the registrant's disclosure controls and
       procedures and presented in this quarterly report our conclusions about
       the effectiveness of the disclosure controls and procedures as of the end
       of the period covered by this quarterly report based on such evaluation;
       and

    c)  disclosed in this quarterly report any change in the registrant's
       internal control over financial reporting that occurred during the
       registrant's most recent fiscal quarter (the registrant's fourth fiscal
       quarter in the case of an annual report) that has materially affected, or
       is reasonably likely to materially affect, the registrant's internal
       control over financial reporting; and

5.  The registrant's other certifying officers and I have disclosed, based on
    our most recent evaluation of internal control over financial reporting, to
    the registrant's auditors and the audit committee of the registrant's board
    of directors (or persons performing the equivalent function):

    a)  all significant deficiencies and material weaknesses in the design or
       operation of internal control over financial reporting which are
       reasonably likely to adversely affect the registrant's ability to record,
       process, summarize and report financial information; and

    b)  any fraud, whether or not material, that involves management or other
       employees who have a significant role in the registrant's internal
       control over financial reporting.

Date: August 10, 2005

<Table>
<S>                                                    <C>  <C>
/s/ JOHN C. MALONE
-------------------------------------------
   John C. Malone
   Chief Executive Officer
</Table>