K2-18 is a nearby M2.5 dwarf, located at 34 pc and hosting a transiting planet that was first discovered by the K2 mission and later confirmed with Spitzer Space Telescope observations. With a radius of ∼2 R⊕ and an orbital period of ∼33 days, the planet lies in the temperate zone of its host star and receives stellar irradiation similar to that of Earth. Here we perform radial velocity follow-up observations with the visual channel of CARMENES with the goal of determining the mass and density of the planet. We measure a planetary semi-amplitude of Kb ∼ 3.5 and a mass of Mb ∼ 9 M⊕, yielding a bulk density around . This indicates a low-mass planet with a composition consistent with a solid core and a volatile-rich envelope. A signal at 9 days was recently reported using radial velocity measurements taken with the HARPS spectrograph. This was interpreted as being due to a second planet. We see a weaker, time- and wavelength-dependent signal in the CARMENES data set and thus favor stellar activity for its origin. K2-18 b joins the growing group of low-mass planets detected in the temperate zone of M dwarfs. The brightness of the host star in the near-infrared makes the system a good target for detailed atmospheric studies with the James Webb Space Telescope.