  We compute exactly the scalar determinants $\det(\Delta+M^{2})$ on the
two-dimensional round disks of constant curvature $R=0$, $\mp 2$, for any
finite boundary length $\ell$ and mass $M$, with Dirichlet boundary conditions,
using the $\zeta$-function prescription. When $M^{2}=\pm q(q+1)$, $q\in\mathbb
N$, a simple expression involving only elementary functions and the Euler
$\Gamma$ function is found. Applications to two-dimensional Liouville and
Jackiw-Teitelboim quantum gravity are presented in a separate paper.
