Let $M$ be a manifold with pinched negative sectional curvature. We show that, when $M$ is geometrically finite and the geodesic flow on $T^1M$ is topologically mixing, the set of mixing invariant measures is dense in the set $P(T^1M)$ of invariant probability measures. This implies that the set of weak-mixing measures which are invariant by the geodesic flow is a dense $G_\delta$ subset of $P(T^1M)$. We also show how to extend these results to geometrically infinite manifolds with cusps or with constant negative curvature.