The diffraction limit prevents a conventional optical microscope from imaging at the nanoscale. However, nanoscale imaging of molecules is possible by exploiting an intensity-dependent molecular switch 1-3 . This switch is translated into a microscopy scheme, stimulated emission depletion microscopy 4-7 . Variants on this scheme exist 3,8-13, yet all exploit an incoherent response to the lasers. We present a scheme that relies on a coherent response to a laser. Quantum control of a two-level system proceeds via rapid adiabatic passage, an ideal molecular switch. We implement this scheme on an ensemble of quantum dots. Each quantum dot results in a bright spot in the image with extent down to 30 nm (λ/31). There is no significant loss of intensity with respect to confocal microscopy, resulting in a factor of 10 improvement in emitter position determination. The experiments establish rapid adiabatic passage as a versatile tool in the super-resolution toolbox.