  To a knot in 3-space, one can associate a sequence of Laurent polynomials,
whose $n$th term is the $n$th colored Jones polynomial. The Volume Conjecture
for small angles states that the value of the $n$-th colored Jones polynomial
at $e^{\a/n}$ is a sequence of complex numbers that grows subexponentially, for
a fixed small complex angle $\a$. In an earlier publication, the authors proved
the Volume Conjecture for small purely imaginary angles, using estimates of the
cyclotomic expansion of a knot. The goal of the present paper is to identify
the polynomial growth rate of the above sequence to all orders with the loop
expansion of the colored Jones function. Among other things, this provides a
strong analytic form of the Melvin-Morton-Rozansky conjecture. The resubmission
corrects a misspelling of the first name of the second author.
