*

Quantum fluctuations in the effective spin-1/2 layered-structure triangular-lattice quantum Heisenberg antiferromagnet Ba₃CoSb₂O₉ lift the classical degeneracy of the antiferromagnetic ground state in magnetic field, producing a series of novel magnetic-field-induced spin structures. We have used specific heat, neutron diffraction, thermal conductivity, and magnetic torque measurements to map out the phase diagram of this system as a function of magnetic field intensity and orientation relative to the crystallographic ab plane. We report previously unobserved magnetic-field-induced phase transitions and a field-orientation dependence that deviates from theoretical predictions near H ‖ c . A tetracritical point in the high field phase diagram and the second-order nature of the transitions eliminates several of the theoretically proposed spin structures for the high field phases and provides new guidance into the nature of the magnetic order in each phase.