On a Riemann surface [`(S)] with smooth boundary, we consider Riemannian metrics conformal to a given background metric. Let κ be a smooth, positive function on [`(S)]. If K denotes the Gauss curvature, then the L ∞-norm of K/κ gives rise to a functional on the space of all admissible metrics. We study minimizers subject to an area constraint. Under suitable conditions, we construct a minimizer with the property that |K|/κ is constant. The sign of K can change, but this happens only on the nodal set of the solution of a linear partial differential equation.