McDiarmid and Reed ['On the maximum degree of a random planar graph', Combin. Probab. Comput. 17 (2008) 591-601] showed that the maximum degree of a random labeled planar graph with vertices satisfies with high probability (w.h.p.)for suitable constants . In this paper, we determine the precise asymptotics, showing in particular that w.h.p. for a constant that we determine explicitly. The proof combines tools from analytic combinatorics and Boltzmann sampling techniques.