We report on novel observations of HAT-P-1 aimed at constraining the optical transmission spectrum of the atmosphere of its transiting Hot-Jupiter exoplanet. Ground-based differential spectrophotometry was performed over two transit windows using the DOLORES spectrograph at the Telescopio Nazionale Galileo (TNG). Our measurements imply an average planet to star radius ratio equal to $rm R_p/R_{star}$=(0.1159$pm$0.0005). This result is consistent with the value obtained from recent near infrared measurements of this object but differs from previously reported optical measurements being lower by around 4.4 exoplanet scale heights. Analyzing the data over 5 different spectral bins 600AA$,$ wide we observed a single peaked spectrum (3.7 $rmsigma$ level) with a blue cut-off corresponding to the blue edge of the broad absorption wing of sodium and an increased absorption in the region in between 6180-7400AA. We also infer that the width of the broad absorption wings due to alkali metals is likely narrower than the one implied by solar abundance clear atmospheric models. We interpret the result as evidence that HAT-P-1b has a partially clear atmosphere at optical wavelengths with a more modest contribution from an optical absorber than previously reported.