Light transmission through Arctic sea ice and snow has an important impact on energy partitioning at the atmosphere-ice-ocean interface and the ice-associated ecosystem. Thus, it is crucial to understand which parameters determine the temporal and spatial variation of sea ice transmission. Ice and snow imprint characteristic features on the spectral shape of transmitted light. Here, we aim to use these spectral features to retrieve snow depth from hyperspectral under-ice light measurements. Transmitted spectral radiance was measured underneath a 100 m long transect on level landfast First-Year-Ice (FYI) using a remotely operated vehicle (ROV). Measurements took place off the northern coast of Ellesmere Island close to the Canadian Forces Station Alert in May 2018. Co-located measurements of ice and snow thickness were acquired with an electromagnetic induction device, a Magna-Probe and a terrestrial laser scanner. The small variation in FYI thickness allows separating the spectral effect of snow depth on transmitted radiance spectra. We retrieve snow depth from spectral transflectance data using an inverse algorithm based on normalized difference indices (NDI). We further fit multiplicative exponential functions to the measured spectra to retrieve wavelength-dependent extinction coefficients of sea ice and snow. Fitted values of broadband bulk extinction coefficients range from 1.8 to 3.5 m-1 for sea ice and from 7.4 to 17.2 m-1 for snow. Mean differences in fitted/calculated and measured modal snow depths are 6 cm for the multiplicative exponential functions and 5 cm using NDIs. 41% of the fitted snow depths lie within 5 cm of the measured snow depths using the multiplicative exponential functions and 42% for the NDI-method. The accuracy of snow depth retrieved from our optical approaches is limited to +/-5 cm, as the variation of snow depth within the optical sensor footprint is between 2.5 and 5.0 cm. Our results show how this optical approach allows to derive large-scale snow depth information from under-ice optical spectra e.g. during autonomous long-range under-ice missions, despite its limited spatial resolution and absolute accuracy.