  In this article we complete our formalism relating the finite-volume energy
spectrum of a scalar quantum field theory to the three-to-three scattering
amplitude, ${\cal M}_3$. In previous work we found a quantization condition
relating the spectrum to a non-standard infinite-volume quantity, denoted
${\cal K}_{{\rm df},3}$. Here we present the relation between ${\cal K}_{{\rm
df},3}$ and ${\cal M}_3$. We then discuss briefly how our now completed
formalism can be practically implemented to extract ${\cal M}_3$ from the
finite-volume energy spectrum.
