  The stabilization theorem for $A$-Hilbert modules was established by G. G.
Kasparov. The equivariant version, in which a locally compact group $H$ acts
properly on a locally compact space $Y$, was proved by N. C. Phillips. This
equivariant theorem involves the Hilbert $(H,C_{0}(Y))$-module
$C_{0}(Y,L^{2}(H)^{\infty})$. It can naturally be interpreted in terms of a
stabilization theorem for proper groupoids, and the paper establishes this
theorem within the general proper groupoid context. The theorem has
applications in equivariant KK-theory and groupoid index theory.
