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The Moon is traditionally thought to have coalesced from the debris ejected by a giant impact onto the early Earth. However, such models struggle to explain the similar isotopic compositions of Earth and lunar rocks at the same time as the system's angular momentum, and the details of potential impact scenarios are hotly debated. Above a high resolution threshold for simulations, we find that giant impacts can immediately place a satellite with similar mass and iron content to the Moon into orbit far outside the Earth's Roche limit. Even satellites that initially pass within the Roche limit can reliably and predictably survive, by being partially stripped then torqued onto wider, stable orbits. Furthermore, the outer layers of these directly formed satellites are molten over cooler interiors and are composed of around 60% proto-Earth material. This could alleviate the tension between the Moon's Earth-like isotopic composition and the different signature expected for the impactor. Immediate formation opens up new options for the Moon's early orbit and evolution, including the possibility of a highly tilted orbit to explain the lunar inclination, and offers a simpler, single-stage scenario for the origin of the Moon.