In this paper, we propose a Gaussian minimum shift keying (GMSK) demodulator structure used in a zero-IF Bluetooth low energy (BLE) transceiver. GMSK is one of the continuous phase modulation (CPM), the primary basis for demodulation is the change of phase. The carrier frequency offset between the transmitter and the receiver affects the phase changes, so a frequency offset estimation and calibration algorithm is presented. The algorithm combines improved one bit differential operation for demodulation and coordinate rotation digital computer (CORDIC) algorithm, which is less complex than the conventional frequency offset estimation algorithm. With a frequency offset within 150 kHz, the signal noise radio (SNR) difference for 0.1 percent bit error rate (BER) demand is 1 dB to the most. This GMSK demodulator is realized in Verilog and validated using FPGA.