Degeneration of an hyperbolic surface of finite volume (compact or non-compact) is precisely defined. Briefly, such a surface is continuously deformed so that the lengths of one or more simple, closed geodesics (called pinching geodesics) approach zero. In the limit, each pinching geodesic corresponds to two cusps on a well-defined limit surface. We prove here that the heat kernel on a degenerating surface converges to that of the limit surface. As a corollary, we obtain that the resolvent kernel also converges for a certain range of its spectral parameter.