Terrestrial Solar System planets either have high mean molecular weight atmospheres, as with Venus, Mars, and Earth, or no atmosphere at all, as with Mercury. We do not have sufficient observational information to know if this is typical of terrestrial planets or a phenomenon unique to the Solar System. The bulk of atmospheric exoplanet studies have focused on hot Jupiters and Neptunes, but recent discoveries of small, rocky exoplanets transiting small, nearby stars provide targets that are amenable to atmospheric study. GJ 1132b has a radius of 1.2 \(R_{\oplus}\) and a mass of 1.6 \(M_{\oplus}\), and orbits an M-dwarf 12 parsecs away from the Solar System. We present results from five transits of GJ 1132b taken with the Magellan Clay Telescope and the LDSS3C multi-object spectrograph. We jointly fit our five data sets when determining the best-fit transit parameters both for the white light curve and wavelength-binned light curves. We bin the light curves into 20 nm wavelength bands to construct the transmission spectrum. Our results disfavor a clear, 10\(\times\) solar metallicity atmosphere at 3.7\(\sigma\) confidence and a 10% H\({}_{2}\)O, 90% H\({}_{2}\) atmosphere at 3.5\(\sigma\) confidence. Our data are consistent with a featureless spectrum, implying that GJ 1132b has a high mean molecular weight atmosphere or no atmosphere at all, though we do not account for the possible presence of aerosols. This result is in agreement with theoretical work which suggests that a planet of GJ 1132b’s mass and insolation should not be able to retain a H\({}_{2}\) envelope.