A dual Banach space X X is Kadec-Klee in the weak * topology if weak * and norm convergence of sequences coincide in the unit sphere of X X . We shall consider a stronger, uniform version of this property. A dual Banach space X X is uniformly Kadec-Klee in the weak * topology (UKK*) if for each ε > 0 \varepsilon > 0 we can find a δ \delta in ( 0 , 1 ) (0,1) such that every weak ∗ * -compact, convex subset C C of the unit ball of X X whose measure of norm compactness exceeds ε \varepsilon must meet the ( 1 − δ ) (1 - \delta ) -ball of X X . We show in this paper that C 1 ( H ) {C_1}(\mathcal {H}) , the space of trace class operators on an arbitrary infinite-dimensional Hilbert space H \mathcal {H} is UKK*. Consequently C 1 ( H ) {C_1}(\mathcal {H}) has weak ∗ * -normal structure. This answers affirmatively a question of A. T. Lau and P. F. Mah. From this it follows that C 1 ( H ) {C_1}(\mathcal {H}) has the weak ∗ * -fixed point property.