Superfluidity in its various forms has been of interest since the observation of frictionless flow in liquid helium II1,2. In three spatial dimensions it is conceptually associated with the emergence of long-range order at a critical temperature. One of the hallmarks of superfluidity, as predicted by the two-fluid model3,4 and observed in both liquid helium5 and in ultracold atomic gases6,7, is the existence of two kinds of sound excitation-the first and second sound. In two-dimensional systems, thermal fluctuations preclude long-range order8,9; however, superfluidity nevertheless emerges at a non-zero critical temperature through the infinite-order Berezinskii-Kosterlitz-Thouless (BKT) transition10,11, which is associated with a universal jump12 in the superfluid density without any discontinuities in the thermodynamic properties of the fluid. BKT superfluids are also predicted to support two sounds, but so far this has not been observed experimentally. Here we observe first and second sound in a homogeneous two-dimensional atomic Bose gas, and use the two temperature-dependent sound speeds to determine the superfluid density of the gas13-16. Our results agree with the predictions of BKT theory, including the prediction of a universal jump in the superfluid density at the critical temperature.