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Orbiting an M dwarf 12 pc away, the transiting exoplanet GJ 1132b is a prime target for transmission spectroscopy. With a mass of 1.7M_⊕and radius of 1.1R_⊕, GJ 1132b’s bulk density indicates that this planet is rocky. Yet with an equilibrium temperature of 580 K, GJ 1132b may still retain some semblance of an atmosphere. Understanding whether this atmosphere exists and its composition will be vital for understanding how the atmospheres of terrestrial planets orbiting M dwarfs evolve. We observe five transits of GJ 1132b with the Wide Field Camera 3 (WFC3) on the Hubble Space Telescope (HST). We find a featureless transmission spectrum from 1.1 to 1.7μm, ruling out cloud-free atmospheres with metallicities <300× solar with >4.8σconfidence. We combine our WFC3 results with transit depths from TESS and archival broadband and spectroscopic observations to find a featureless spectrum across 0.7 to 4.5μm. GJ 1132b therefore has a high mean molecular weight atmosphere, possesses a high-altitude aerosol layer, or has effectively no atmosphere. Higher-precision observations are required in order to differentiate between these possibilities. We explore the impact of hot and cold starspots on the observed transmission spectrum GJ 1132b, quantifying the amplitude of spot-induced transit depth features. Using a simple Poisson model, we estimate spot temperature contrasts, spot covering fractions, and spot sizes for GJ 1132. These limits, as well as the modeling framework, may be useful for future observations of GJ 1132b or other planets transiting similarly inactive M dwarfs.