Adaptive identification of autoregressive (AR) signals subject to white measurement noise is studied. A fast adaptive algorithm, which is based on the recently proposed improved least-squares (LS) method, is developed. The variance of the white measurements noise, which specifies the source of the noise-induced bias in the standard LS estimate, is calculated by means of extra noisy measurements of the AR signal. With a good estimate of the measurement noise variance being attained more quickly, the convergence speed of the developed adaptive identification algorithm can be accelerated. Numerical results are presented to demonstrate the promising performance of the new fast adaptive identification algorithm.