According to random-matrix theory, interference effects in the conductance of a ballistic chaotic quantum dot should vanish propto(tau_phi/tau_D)^p when the dephasing time tau_phi becomes small compared to the mean dwell time tau_D. Aleiner and Larkin have predicted that the power law crosses over to an exponential suppression exp(�tau_E/tau_phi) when tau_phi drops below the Ehrenfest time tau_E. We report the first observation of this crossover in a computer simulation of universal conductance fluctuations. Their theory also predicts an exponential suppression propto exp(�tau_E/tau_D) in the absence of dephasing�which is not observed. We show that the effective random-matrix theory proposed previously for quantum dots without dephasing explains both observations.