The problem of blind separation of underdetermined instantaneous mixtures of independent signals is addressed through a method relying on nonstationarity of the original signals. The signals are assumed to be piecewise stationary with varying variances in different epochs. In comparison with previous works, in this paper it is assumed that the signals are not i.i.d. in each epoch, but obey a first-order autoregressive model. This model was shown to be more appropriate for blind separation of natural speech signals. A separation method is proposed that is nearly statistically efficient (approaching the corresponding Cramér-Rao lower bound), if the separated signals obey the assumed model. In the case of natural speech signals, the method is shown to have separation accuracy better than the state-of-the-art methods.