The stresses induced in the vacuum by the uniform acceleration of an infinite plane conductor are computed for the massless scalar and electromagnetic fields. Both Dirichlet and Neumann boundary conditions are considered for the scalar field; far from the conductor it is found, independently of the boundary condition, that the vacuum stress is 'local' and corresponds to the absence from the vacuum of black body radiation. Approaching the conductor, the energy density in the Dirichlet case is slightly lower than the 'local' term, and in the Neumann case slightly higher. At very small distances it again has the same asymptotic form for both scalar fields. For the electromagnetic field the results are similar to those for the scalar field with Dirichlet boundary conditions. Far from the conductor the spectrum is again black-body, though not Planckian. In all cases the acausal nature of 'perfect conductor' boundary conditions prevents the stress tensor from being finite on the conductor.