We consider the contact process on a random graph with a fixed degree distribution given by a power law. We follow the work of Chatterjee and Durrett [2], 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. For lambda close to 0 (that is, "near criticality"), we obtain sharp bounds for the typical density of infected sites in the graph, as the number of vertices tends to infinity. We exhibit three different regimes for this density, depending on the tail of the degree law.