We introduce and analyze a new mixed finite element method for the spatial discretization of an incompressible magnetohydrodynamics problem. It is based on divergence-conforming elements for the fluid velocities and on curl-conforming elements for the magnetic unknowns. The tangential continuity of the velocities is enforced by a DG approach. Central features of the resulting method are that it produces exactly divergence-free velocity approximations and is provably energy-stable, and that it correctly captures the strongest magnetic singularities in non-smooth domains. We carry out the error analysis of the method, and present a comprehensive set of numerical tests in two and three dimensions. We also discuss some recent ideas regarding the design of efficient solvers for the matrix systems.