  Let $h$ be a complete metric of Gaussian curvature $K_0$ on a punctured
Riemann surface of genus $g \geq 1$ (or the sphere with at least three
punctures). Given a smooth negative function $K$ with $K=K_0$ in neighbourhoods
of the punctures we prove that there exists a metric conformal to $h$ which
attains this function as its Gaussian curvature for the punctured Riemann
surface. We do so by minimizing an appropriate functional using elementary
analysis.
