<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.

<Table>
<S>                                                       <C>
Date: November 9, 2005

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