We propose a finite element method for the three-dimensional transient incompressible magnetohydrodynamic equations that ensures exactly divergence-free approximations of the velocity and the magnetic induction. We employ second-order semi-implicit timestepping, for which we rigorously establish an energy law and, as a consequence, unconditional stability. We prove unique solvability of the linear systems of equations to be solved in every timestep. For those we design an efficient preconditioner so that the number of preconditioned GMRES iterations is uniformly bounded with respect to the number of degrees of freedom As both meshwidth and timestep size tend to zero, we prove that the discrete solutions converge to a weak solution of the continuous problem. Finally, by several numerical experiments, we confirm the predictions of the theory and demonstrate the efficiency of the preconditioner.