  We consider the contact process on a random graph with fixed degree
distribution given by a power law. We follow the work of Chatterjee and
Durrett, who showed that for arbitrarily small infection parameter $\lambda$,
the survival time of the process is larger than a stretched exponential
function of the number of vertices, $n$. We obtain sharp bounds for the typical
density of infected sites in the graph, as $\lambda$ is kept fixed and $n$
tends to infinity. We exhibit three different regimes for this density,
depending on the tail of the degree law.
