No process in nature can perfectly copy an arbitrary quantum state. But is it possible to engineer processes that replicate quantum states with vanishingly small error? We devised a probabilistic process that transforms N photons with the same linear polarization into M photons polarized approximately in the same direction, with an error that vanishes exponentially fast whenever M is small compared to the square of N. To explain the roots of this super-cloning phenomenon, we establish a fundamental connection between quantum cloning and the precision limits of quantum metrology, showing that the Heisenberg limit determines the number of clones that can be produced with vanishing error by a general physical process.