Barnard's star is a red dwarf, and has the largest proper motion (apparent motion across the sky) of all known stars. At a distance of 1.8 parsecs(1), it is the closest single star to the Sun; only the three stars in the alpha Centauri system are closer. Barnard's star is also among the least magnetically active red dwarfs known(2,3) and has an estimated age older than the Solar System. Its properties make it a prime target for planetary searches; various techniques with different sensitivity limits have been used previously, including radial-velocity imagine(4-6), astrometry(7,8 )and direct imaging(9), but all ultimately led to negative or null results. Here we combine numerous measurements from high-precision radial-velocity instruments, revealing the presence of a low-amplitude periodic signal with a period of 233 days. Independent photometric and spectroscopic monitoring, as well as an analysis of instrumental systematic effects, suggest that this signal is best explained as arising from a planetary companion. The candidate planet around Barnard's star is a cold super-Earth, with a minimum mass of 3.2 times that of Earth, orbiting near its snow line (the minimum distance from the star at which volatile compounds could condense). The combination of all radial-velocity datasets spanning 20 years of measurements additionally reveals a long-term modulation that could arise from a stellar magnetic-activity cycle or from a more distant planetary object. Because of its proximity to the Sun, the candidate planet has a maximum angular separation of 220 milliarcseconds from Barnard's star, making it an excellent target for direct imaging and astrometric observations in the future.