Measurements of the superfluid density and heat capacity in porous aerogel glasses that are completely filled with liquid helium are presented as examples of critical behavior in the presence of a dilute correlated impurity. Longstanding doubts about the existence of critical behavior in filled4 porous media, brought about by the failure of all previous searches for a singularity in anything other than the superfluid density, are put to rest by the observation of a sharp heat-capacity singularity that is coincident with the superfluid transition. With the notable exception of the superfluid transition in pure helium itself, the superfluid transitions in filled4 aerogel are among the sharpest phase transitions ever seen. However, the critical exponents for the superfluid density and heat capacity are both markedly different from the corresponding exponents in pure helium. We conclude that the correlated disorder introduced by the aerogel glass drives this system away from the three-dimensional XY universality class to which pure helium belongs.