Observations of the Kepler-1625 system with the Kepler and Hubble Space Telescopes have suggested the presence of a candidate exomoon, Kepler-1625b I, a Neptune-radius satellite orbiting a long-period Jovian planet. Here we present a new analysis of the Hubble observations, using an independent data reduction pipeline. We find that the transit light curve is well fit with a planet-only model, with a best-fit $chi^2_ u$ equal to 1.01. The addition of a moon does not significantly improve the fit quality. We compare our results directly with the original light curve from Teachey & Kipping (2018), and find that we obtain a better fit to the data using a model with fewer free parameters (no moon). We discuss possible sources for the discrepancy in our results, and conclude that the lunar transit signal found by Teachey & Kipping (2018) was likely an artifact of the data reduction. This finding highlights the need to develop independent pipelines to confirm results that push the limits of measurement precision.