Time-series spectrophotometric studies of exoplanets during transit using ground-based facilities are a promising approach to characterize their atmospheric compositions. We aim to investigate the transit spectrum of the hot Jupiter HAT-P-1b. We compare our results to those obtained at similar wavelengths by previous space-based observations. We observed two transits of HAT-P-1b with the Gemini Multi-Object Spectrograph (GMOS) instrument on the Gemini North telescope using two instrument modes covering the 320-800nm and 520-950nm wavelength ranges. We used time-series spectrophotometry to construct transit light curves in individual wavelength bins and measure the transit depths in each bin. We accounted for systematic effects. We addressed potential photometric variability due to magnetic spots in the planet's host star with long-term photometric monitoring. We find that the resulting transit spectrum is consistent with previous Hubble Space Telescope (HST) observations. We compare our observations to transit spectroscopy models that marginally favor a clear atmosphere. However, the observations are also consistent with a flat spectrum, indicating high-altitude clouds. We do not detect the Na resonance absorption line (589nm), and our observations do not have sufficient precision to study the resonance line of K at 770nm. We show that even a single Gemini/GMOS transit can provide constraining power on the properties of the atmosphere of HAT-P-1b to a level comparable to that of HST transit studies in the optical when the observing conditions and target and reference star combination are suitable. Our 520-950nm observations reach a precision comparable to that of HST transit spectra in a similar wavelength range of the same hot Jupiter, HAT-P-1b. However, our GMOS transit between 320-800nm suffers from strong systematic effects and yields larger uncertainties.