  We consider the perturbations of the massive vector field around
Schwarzschild black hole, (generally, with non-vanishing $\Lambda$ - term). The
monopole massive vector perturbation equations can be reduced to a single
wave-like equation. We have proved the stability against these perturbations
and investigated the quasinormal spectrum. The quasinormal behaviour for
Schwarzschild black hole is quite unexpected: the fundamental mode and all
higher overtones shows totally different dependence on the mass of the field
$m$: as $m$ is increasing, the damping rate of the fundamental mode is
decreasing, what results in appearing of the infinitely long living modes,
while, on contrary, damping rate of all higher overtones are increasing, and
their real oscillation frequencies gradually go to tiny values. Thereby, for
all higher overtones, almost non-oscillatory, damping modes can exist. In the
limit of asymptotically high damping, $Re \omega$ goes to $ln3/(8 \pi M)$,
while imaginary part shows equidistant behaviour with spacing $Im \omega_{n+1}-
Im \omega_{n}=i/4M$. In addition, we have found quasinormal spectrum of massive
vector field for Schwarzschild-anti-de Sitter black hole.
