On a Spector Ultrapower for the Solovay Model

We prove that a Spector-like ultrapower extension ?? of a countable Solovay model ?? (where all sets of reals are Lebesgue measurable) is equal to the set of all sets constructible from reals in a generic extension ??a], where a is a random real over ??. The proof involves the Solovay almost everywhere uniformization technique.