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Efforts to identify habitable extrasolar planets have focused on systems around M dwarfs, faint stars with less than half the solar mass. Habitable planets around M dwarfs are thought to be more plentiful and easier to detect than those orbiting Sun-like G dwarfs(1-4). However, unlike G dwarfs, M dwarfs experience a prolonged decline in luminosity early in their history, leading to an inward migration of the habitable zone to where planets may have lost their water through dissociation and hydrodynamic escape. Water-poor planets, such as Venus, are considered uninhabitable. In contrast, planets with too much water (>1 wt%) would lack continents(5), leading to climate instability(6) and nutrient limitation problems(7). Here we combine a numerical planet population synthesis model with a model for water loss to show that the evolution of stellar luminosity leads to two types of planets of Earth-like mass (0.1 to 10 Earth masses) in the habitable zones around M dwarfs: ocean planets without continents, and desert planets, on which there are orders of magnitude less surface water than on Earth. According to our simulations, Earth-mass planets with Earth-like water contents are rare around M dwarfs and occur 10-100 times less frequently than around G dwarfs. We suggest that stars close to the size of the Sun should be the primary targets for detecting Earth-like planets.