We prove an analogue, for surfaces with constant mean curvature in hyperbolic space, of a theorem of Fischer-Colbrie and Gulliver about minimal surfaces in Euclidean space. That is, for a complete surface
M
2
{M^2}
in hyperbolic
3
3
-space with constant mean curvature 1, the (Morse) index of the operator
L
=
Δ
−
2
K
L = \Delta - 2K
is finite if and only if the total Gaussian curvature is finite.