Let B be a separable Banach space and let X = B* be separable. We prove that if B has finite Szlenk index (for all ε > 0) then B can be renormed to have the weak* uniform Kadec-Klee property. Thus if ε > 0 there exists δ(ε) > 0 so that if (xn) is a sequence in the ball of X converging ω* to x so that lim infn→∞ ||xn - x|| ≥ ε then ||x|| ≤ 1 - δ(ε). In addition we show that the norm can be chosen so that δ(ε) ≥ cεp for some p < ∞ and c > 0. © 1999 Kluwer Academic Publishers.