Mars today is much drier than the Earth, though they likely began with similar relative amounts of water. One potential cause for this discrepancy is hydrogen loss to space, which may have removed a large fraction of Mars' initial water. Here we demonstrate an order-of-magnitude change in the Martian hydrogen escape rate in 2007, inconsistent with established models for the source of escaping hydrogen. We analyze 121.6 nm (hydrogen Lyman-α) airglow observations made by the ultraviolet spectrometer on the Mars Express spacecraft over the second half of 2007. The enhanced escape rates we observe may be due to lower atmospheric heating and overturn during the 2007 (Mars Year 28) global dust storm, suggesting that hydrogen escape from Mars during dust storms may dominate loss of the planet's water inventory. This scenario has major implications for reconstructing the total amount of water lost to space over Martian history.