Following Dungey's original magnetohydrodynamic formulation, a solution is obtained for the nonlinear evolution of a current discharge in the vicinity of a magnetic neutral line. For an ideal gas with constant conductivity and uniform mass density we obtain a particular exact solution in the limit of an initial, nearly sheet-like configuration. This particular solution implies special boundary conditions for the pressure and electric field at the surface of the conductor. If These conditions are not met, then the solution eventually breaks down before the current density becomes infinite. The time required for complete breakdown is determined by the wave propagation times from the surface of the fluid to the neutral line and by the diffusion time for the magnetic field through the fluid. For large conductivity and a small sound speed, the maximum current density achieved at the time of the solution's complete breakdown depends upon the ratio of the characteristic diffusion time to the Alfvén wave propagation time. In the limit of infinite conductivity or infinite extension of the fluid, the current density along the neutral line becomes infinite at π/23/2 times the Alfvénic scale time. At this same time the inflow Alfvenic Mach number approaches 21/2i while the outflow Alfvenic Mach number approaches infinity.