We present some new results concerning the Dirichlet problem for the prescribed mean curvature equation over a bounded domain in R^n. In the case when the mean curvature is zero this can be posed variationally as the problem of finding a least area representative among functions of bounded variation with prescribed boundary values. We show that there is always a minimizer which is represented by a compact C^{1,alpha} manifold with boundary, with boundary given by the prescribed Dirichlet data, provided this data is C^{1,alpha} and it is of class C^{1,1} if the prescribed data is C^3.