  We prove: "If $M$ is a compact hypersurface of the hyperbolic space, convex
by horospheres and evolving by the volume preserving mean curvature flow, then
it flows for all time, convexity by horospheres is preserved and the flow
converges, exponentially, to a geodesic sphere". In addition, we show that the
same conclusions about long time existence and convergence hold if $M$ is not
convex by horospheres but it is close enough to a geodesic sphere.
