We present an algorithm for learning a mixture of distributions based on spectral projection. We prove a general property of spectral projection for arbitrary mixtures and show that the resulting algorithm is efficient when the components of the mixture are logconcave distributions in $\Re^{n}$ whose means are separated. The separation required grows with k, the number of components, and log n. This is the first result demonstrating the benefit of spectral projection for general Gaussians and widens the scope of this method. It improves substantially on previous results, which focus either on the special case of spherical Gaussians or require a separation that has a considerably larger dependence on n.