We present a new method for probabilistically classifying supernovae (SNe) without using SN spectral or photometric data. Unlike all previous studies to classify SNe without spectra, this technique does not use any SN photometry. Instead, the method relies on host-galaxy data. We build upon the well-known correlations between SN classes and host-galaxy properties, specifically that core-collapse SNe rarely occur in red, luminous, or early-type galaxies. Using the nearly spectroscopically complete Lick Observatory Supernova Search sample of SNe, we determine SN fractions as a function of host-galaxy properties. Using these data as inputs, we construct a Bayesian method for determining the probability that a SN is of a particular class. This method improves a common classification figure of merit by a factor of >2, comparable to the best light-curve classification techniques. Of the galaxy properties examined, morphology provides the most discriminating information. We further validate this method using SN samples from the Sloan Digital Sky Survey and the Palomar Transient Factory. We demonstrate that this method has wide-ranging applications, including separating different subclasses of SNe and determining the probability that a SN is of a particular class before photometry or even spectra can. Since this method uses completely independent data from light-curve techniques, there is potential to further improve the overall purity and completeness of SN samples and to test systematic biases of the light-curve techniques. Further enhancements to the host-galaxy method, including additional host-galaxy properties, combination with light-curve methods, and hybrid methods should further improve the quality of SN samples from past, current, and future transient surveys.