Hydrogen bonding between water molecules and zeolite Bro̷nsted acid sites (BAS) has received much attention due to the significant influence of water on the adsorption and catalytic properties of these widely used porous materials. When a single water molecule is adsorbed at the BAS, the zeolite O-H stretch vibration decreases in frequency and splits into two extraordinarily broad bands peaked near 2500 and 2900 cm-1 in the infrared (IR) spectrum. This broad doublet feature is the predominant IR signature used to identify and interpret water-BAS H-bonding at low hydration levels, but the origin of the band splitting is not well understood. In this study, we used broadband two-dimensional infrared (2D IR) spectroscopy to investigate zeolite HZSM-5 prepared with a single water molecule per BAS. We find that the 2D IR spectrum is not explained by the most common interpretation of Fermi resonance coupling between the stretch and the bend of the BAS OH group, which predicts intense excited-state transitions that are absent from the experimental results. We present an alternative model of a double-well proton stretch potential, where the band splitting is caused by excited-state tunneling through the proton-transfer barrier. This one-dimensional model reproduces the basic experimental pattern of transition frequencies and amplitudes, suggesting that the doublet bands may originate from a highly anharmonic potential in which the excited state proton wave functions are delocalized over the H-bond between zeolite BAS and adsorbed H2O. Additional details about molecular orientation and coordination of the adsorbed water molecule are also resolved in the 2D IR spectroscopy.