Context. The origin of water, particularly on Earth, is still a matter of heated debate. To date, the most widespread scenario is that the Earth originated without water and that it was brought to the planet mainly as a result of impacts by wet asteroids coming from further out in space. However, many uncertainties remain as to the exact processes that supplied an adequate amount of water to inner terrestrial planets.Aims. In this article, we explore a new mechanism that would allow water to be efficiently transported to planets without impacts. We propose that primordial asteroids were icy and that when the ice sublimated, it formed a gaseous disk that could then reach planets and deliver water.Methods. We have developed a new model that follows the sublimation of asteroids on gigayear (Gyr) timescales, taking into account the variable luminosity of the Sun. We then evolved the subsequent gas disk using a viscous diffusion code, which leads to the gas spreading both inwards and outwards in the Solar System. We can then quantify the amount of water that can be accreted onto each planet in a self-consistent manner using our code.Results. We find that this new disk-delivery mechanism is effective and equipped to explain the water content on Earth (with the correct D/H ratio) as well as on other planets and the Moon. Our model shows most of the water being delivered between 20 and 30 Myr after the birth of the Sun, when the Sun’s luminosity increased sharply. Our scenario implies the presence of a gaseous water disk with substantial mass for hundreds of millions of years, which could be one of the key tracers of this mechanism. We show that such a watery disk could be detected in young exo-asteroid belts with ALMA.Conclusions. We propose that viscous water transport is inevitable and more generic than the impact scenario. We also suggest it is a universal process that may also occur in extrasolar systems. The conditions required for this scenario to unfold are indeed expected to be present in most planetary systems: an opaque proto-planetary disk that is initially cold enough for ice to form in the exo-asteroid belt region, followed by a natural outward-moving snow line that allows this initial ice to sublimate after the dissipation of the primordial disk, creating a viscous secondary gas disk and leading to the accretion of water onto the exo-planets.