The asymptotic normality of a class of estimators for extreme quantiles is established under mild structural conditions on the observed stationary β-mixing time series. Consistent estimators of the asymptotic variance are introduced, which render possible the construction of asymptotic confidence intervals for the extreme quantiles. Moreover, it is shown that many well-known time series models satisfy our conditions. The theory is then applied to a time series of stock index returns. Finally, the finite-sample behaviour of the proposed confidence intervals is examined in a simulation study. It turns out that for most time series models under consideration the actual coverage probability is pretty close to the nominal level if the sample fraction used for estimation is chosen appropriately.