Kahler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2 \pi

This is the second of a series of three papers which provide proofs of results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle is less than 2\pi. We show that these are in a natrual way projective algebraic varieties. In the case when the limiting variety and the limiting divisor are smooth we show that the limiting metric also has standard cone singularities.