Convection in a vertical magnetic field occurs in narrow cells in the physically relevant limit where the Chandrasekhar number Q becomes large, corresponding to a strong field or small diffusion. This allows asymptotic solutions to be developed for fully nonlinear convection, requiring only the solution of a nonlinear boundary value problem. Solutions for steady and oscillatory magnetoconvection are obtained, with different scalings. In the steady case, the heat flux and the fluid velocity are found at leading order in the asymptotic expansion and the vertical velocity scales as Q1/6. In the oscillatory case, where it is necessary to continue to second order, the vertical velocity is of order Q1/3 and the frequency of the oscillations is always greater than that predicted by linear theory. The heat flux does not depend on either the wavenumber or the planform.