At zero temperature and strong applied magnetic fields the ground state of an anisotropic antiferromagnet is a saturated paramagnet with fully aligned spins. We study the quantum phase transition as the field is reduced below an upper critical Hc2 and the system enters a XY -antiferromagnetic phase. Using a bond operator representation we consider a model spin-1 Heisenberg antiferromagnetic with single-ion anisotropy in hypercubic lattices under strong magnetic fields. We show that the transition at Hc2 can be interpreted as a Bose-Einstein condensation (BEC) of magnons. The theoretical results are used to analyze our magnetization versus field data in the organic compound Ni Cl2 -4SC (N H2) 2 (DTN) at very low temperatures. This is the ideal BEC system to study this transition since Hc2 is sufficiently low to be reached with static magnetic fields (as opposed to pulsed fields). The scaling of the magnetization as a function of field and temperature close to Hc2 shows excellent agreement with the theoretical predictions. It allows us to obtain the quantum critical exponents and confirm the BEC nature of the transition at Hc2. © 2008 The American Physical Society.