  We develop a finite element method for the Laplace-Beltrami operator on a
surface with boundary and nonhomogeneous Dirichlet boundary conditions. The
method is based on a triangulation of the surface and the boundary conditions
are enforced weakly using Nitsche's method. We prove optimal order a priori
error estimates for piecewise continuous polynomials of order $k \geq 1$ in the
energy and $L^2$ norms that take the approximation of the surface and the
boundary into account.
