We study the phase transition of Bose—Einstein condensation in a dilute atomic gas very close to the critical temperature. The critical regime we enter is governed by fluctuations extending far beyond the length scale of thermal de-Broglie waves. Using matter-wave interference we measure the correlation length of these critical fluctuations as a function of temperature. From this we determine the critical exponent of the correlation length for a trapped, weakly interacting Bose gas to be v = 0.67 ± 0.13.