Fast radio bursts (FRBs) are bright, millisecond-duration radio transients originating from extragalactic distances. Their origin is unknown. Some FRB sources emit repeat bursts, ruling out cataclysmic origins for those events. Despite searches for periodicity in repeat burst arrival times on time scales from milliseconds to many days, these bursts have hitherto been observed to appear sporadically, and though clustered, without a regular pattern. Here we report the detection of a 16.35±0.18 day periodicity from a repeating FRB 180916.J0158+65 detected by the Canadian Hydrogen Intensity Mapping Experiment Fast Radio Burst Project (CHIME/FRB). In 28 bursts recorded from 16th September 2018 through 30th October 2019, we find that bursts arrive in a 4.0-day phase window, with some cycles showing no bursts, and some showing multiple bursts, within CHIME’s limited daily exposure. Our results suggest a mechanism for periodic modulation either of the burst emission itself, or through external amplification or absorption, and disfavour models invoking purely sporadic processes.