We establish a nonlinear Korn inequality with boundary conditions showing that the H¹-distance between two mappings from Ω⊂ℝⁿ into ℝⁿ preserving orientation is bounded, up to a multiplicative constant, by the L²-distance between their metrics. This inequality is then used to show the existence of a unique minimizer to the total energy of a hyperelastic body, under the assumptions that the L^p-norm of the density of the applied forces is small enough and the stored energy function is bounded from below by a positive definite quadratic function of the Green-Saint Venant strain tensor.