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Active front end (AFE) rectifiers are introduced with bidirectional power transfer capability for the power factor correction and current harmonics elimination. This paper proposes an adaptive model predictive control (MPC) for a single-phase AFE rectifier. The AFE topology used in this paper, is a half-bridge boost converter. In comparison with other single-phase AFEs, this topology has higher efficiency due to a lower number of active semiconductor devices. Considering the presence of one leg of capacitors in this topology, the balancing of the these capacitors voltages is known to be a challenge, along with other control objectives. Also, this paper benefits from an MPC control approach by which the switching signal for every switching state is chosen such that the defined cost function becomes minimized in operation state to achieve all control goals based on the predefined model of the rectifier. Furthermore, an adaptive algorithm for the inductor current estimation is presented to robust the control system against any unwanted disturbance in the control system. As a demonstration of the superior performance of the proposed method, a 1kW rectifier setup with 700Vdc output voltage fed from a 230Vrms/50Hz grid is built in the laboratory. The applicability of the MPC controller is verified and all operation waveforms of the test setup are presented. The proposed method offers a high-quality input current with total harmonic distortion (THD) below 5% and a high power factor (PF) close to one. Also, the efficiency is comparable with the available commercialized rectifiers