  If a hyperbolic link has a prime alternating diagram D, then we show that the
link complement's volume can be estimated directly from D. We define a very
elementary invariant of the diagram D, its twist number t(D), and show that the
volume lies between v_3(t(D) - 2)/2 and v_3(16t(D) - 16), where v_3 is the
volume of a regular hyperbolic ideal 3-simplex. As a consequence, the set of
all hyperbolic alternating and augmented alternating link complements is a
closed subset of the space of all complete finite volume hyperbolic
3-manifolds, in the geometric topology. The appendix by Ian Agol and Dylan
Thurston, which was written after the first version of this paper was
distributed, improves the upper bound on volume to v_3(10t(D) - 10). In
addition, examples of alternating links are given which asymptotically achieve
this bound.
