We demonstrate in-vacuum, magnetic levitation of a compact high-temperature superconducting (HTS) coil, which has been designed for magnetic confinement of an electron-positron pair plasma. The closed, no-insulation (NI) rare-earth barium copper oxide (ReBCO) coil was energized with a persistent current to generate a dipole magnetic field, then magnetically levitated by a water-cooled copper lifting coil located above. The vertical position of the floating coil was measured by an array of laser position sensors. Stable levitation was achieved by continuous adjustment of the lifting coil current using a 1 kHz Proportional-Integral-Derivative (PID) feedback loop implemented by a Field-Programmable Gate Array (FPGA). The feedback parameters were optimized with a 1D simulation of the levitation system. A levitation time in excess of 3 hours was achieved with a mean vertical displacement from the setpoint position of -3 µm and a standard deviation of σ_z = 18 µm.