Humans have the remarkable ability to continually store new memories, while maintaining old memories for a lifetime. How the brain avoids catastrophic forgetting of memories due to interference between encoded memories is an open problem in computational neuroscience. Here we present a model for continual learning in a recurrent neural network combining Hebbian learning, synaptic decay and a novel memory consolidation mechanism: memories undergo stochastic rehearsals with rates proportional to the memory’s basin of attraction, causing self-amplified consolidation. This mechanism gives rise to memory lifetimes that extend much longer than the synaptic decay time, and retrieval probability of memories that gracefully decays with their age. The number of retrievable memories is proportional to a power of the number of neurons. Perturbations to the circuit model cause temporally-graded retrograde and anterograde deficits, mimicking observed memory impairments following neurological trauma.