  We give upper bounds for the determinant of an $n\times n$ zero-one matrix
containing $kn$ ones for integral $k$. Our results improve upon a result of
Ryser for $k=o(n^{1/3})$. For fixed $k\ge 3$ it was an open question whether
Hadamard's inequality could be exponentially improved. We answer this in the
affirmative. Our results stem from studying matrices with row sums $k$ and
bounding their Gram determinants. Our technique allows us to give upper bounds
when these matrices are perturbed.
