  We prove, assuming that the conjecture of Lang and Vojta holds true, that
there is a uniform bound on the number of stably integral points in the
complement of the theta divisor on a principally polarized abelian surface
defined over a number field. This gives a uniform version, in the spirit of a
result of Caporaso-Harris-Mazur, of an unconditional theorem of Faltings. We
utilize recent results of Alexeev and Nakamura on complete moduli for
quasi-abelian varieties. We expect that a thorough understanding of current
work of Alexeev should give a more general result for abelian varieties of an
arbitrary dimension with a polarizing divisor of an arbitrary degree - a
proposed approach for such a generalization is given at the end of the paper.
