[en] We report on a Monte Carlo study of two-dimensional Ginzburg-Landau superconductors in a magnetic field. We find evidence suggestive of a first-order phase transition between a low-temperature state characterized by power-law positional correlations in the superfluid density and a high-temperature vortex-fluid state. A key aspect of our study is the introduction of a quantity proportional to the Fourier transform of the superfluid density which can be sampled efficiently in Landau gauge Monte Carlo simulations and which satisfies a useful sum rule