  We prove that a Spector--like ultrapower extension $\gN$ of a countable
Solovay model $\gM$ (where all sets of reals are Lebesgue measurable) is equal
to the set of all sets constructible from reals in a generic extension
$\gM[\al]$ where $\al$ is a random real over $\gM.$ The proof involves an
almost everywhere uniformization theorem in the Solovay model.
