We show that gravitational time dilation constrains the 00 components of metrics of static gravitational fields of finite extent to be nonnegative. Since this constraint is violated by the textbook "Schwarzschild metric" of a static point mass (which is due to Droste and Hilbert, not Schwarzschild), that metric must be reconsidered. It in fact is only one member of a one-parameter family of similar metrics which all meet the requirements set out in textbooks. Although those requirements don't include a boundary condition at the location of the static point mass, gravitational time dilation causes the 00 component of the metric to vanish there. The consequent unique metric of a static point mass turns out to be precisely the metric given in Karl Schwarzschild's January 13, 1916 paper. Schwarzschild's paper, however, doesn't take gravitational time dilation into account; it instead improperly attaches physical significance to a singularity in a factor of a metric component which itself is nonsingular.