A quantum absorption refrigerator driven by noise is studied with the purpose of determining the limitations of cooling to absolute zero. The model consists of a working medium coupled simultaneously to hot, cold, and noise baths. Explicit expressions for the cooling power are obtained for Gaussian and Poisson white noise. The quantum model is consistent with the first and second laws of thermodynamics. The third law is quantified; the cooling power J c vanishes as J cTcα, when T c→0, where α=d+1 for dissipation by emission and absorption of quanta described by a linear coupling to a thermal bosonic field, where d is the dimension of the bath.