We construct a determinant of the Laplacian for infinite-area
surfaces that are hyperbolic near $\infty$ and without cusps. In the
case of a convex cocompact hyperbolic metric, the determinant can be
related to the Selberg zeta function and thus shown to be an entire
function of order 2 with zeros at the eigenvalues and resonances of
the Laplacian. In the hyperbolic near $\infty$ case, the determinant
is analyzed through the zeta-regularized relative determinant for a
conformal metric perturbation. We establish that this relative
determinant is a ratio of entire functions of order 2 with divisor
corresponding to eigenvalues and resonances of the perturbed and
unperturbed metrics. These results are applied to the problem of
compactness in the smooth topology for the class of metrics with a
given set of eigenvalues and resonances.