  We present a systematic method to extract the entire tree-level S-matrix on
the Coulomb branch of N=4 SYM from soft-scalar limits of on-shell amplitudes at
the origin of moduli space. Massive amplitudes in the spontaneously-broken
theory can thus be computed from on-shell amplitudes in the massless, unbroken
theory. To check this correspondence, we first prove that soft and collinear
divergences in the required massless amplitudes cancel for a judicious choice
of soft-scalar momenta. We then explicitly verify our proposal in examples with
arbitrarily many external legs and to all orders in the mass. As a byproduct,
the construction leads to a massive CSW-like expansion that reproduces several
known all-n results for Coulomb-branch amplitudes in an effortless way. We
briefly discuss the extension of our method to loop integrands.
