Mammals and birds engage in two distinct states of sleep, slow wave sleep (SWS) and rapid eye movement (REM) sleep. SWS is characterized by slow, high amplitude brain waves, while REM sleep is characterized by fast, low amplitude waves, known as activation, occurring with rapid eye movements and reduced muscle tone. However, monotremes (platypuses and echidnas), the most basal (or 'ancient') group of living mammals, show only a single sleep state that combines elements of SWS and REM sleep, suggesting that these states became temporally segregated in the common ancestor to marsupial and eutherian mammals. Whether sleep in basal birds resembles that of monotremes or other mammals and birds is unknown. Here, we provide the first description of brain activity during sleep in ostriches (Struthio camelus), a member of the most basal group of living birds. We found that the brain activity of sleeping ostriches is unique. Episodes of REM sleep were delineated by rapid eye movements, reduced muscle tone, and head movements, similar to those observed in other birds and mammals engaged in REM sleep; however, during REM sleep in ostriches, forebrain activity would flip between REM sleep-like activation and SWS-like slow waves, the latter reminiscent of sleep in the platypus. Moreover, the amount of REM sleep in ostriches is greater than in any other bird, just as in platypuses, which have more REM sleep than other mammals. These findings reveal a recurring sequence of steps in the evolution of sleep in which SWS and REM sleep arose from a single heterogeneous state that became temporally segregated into two distinct states. This common trajectory suggests that forebrain activation during REM sleep is an evolutionarily new feature, presumably involved in performing new sleep functions not found in more basal animals.