Summary: In a graph \(G\) with a given edge colouring, a rainbow path is a path all of whose edges have distinct colours. The minimum number of colours required to colour the edges of \(G\) so that every pair of vertices is joined by at least one rainbow path is called the rainbow connection number \(\mathrm{rc}(G)\) of the graph \(G\). For any graph \(G\), \(\mathrm{rc}(G) \geqslant \mathrm{diam}(G)\). We will show that for the Erdős-Rényi random graph \(\mathcal{G}(n,p)\) close to the diameter 2 threshold, with high probability if \(\mathrm{diam}(G)=2\) then \(\mathrm{rc}(G)=2\). In fact, further strengthening this result, we will show that in the random graph process, with high probability the hitting times of diameter 2 and of rainbow connection number 2 coincide.