The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is considered. In the previous works [Moiseev and Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009004; Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009031], extremal trajectories were defined, their local and global optimality were studied. In this paper the global structure of the exponential mapping is described. On this basis an explicit characterization of the cut locus and Maxwell set is obtained. The optimal synthesis is constructed.