  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.
