We report results of dielectric spectroscopy on water confined in a two-dimensional layer-structured Na-vermiculite clay. Several relaxation processes of different origin can be discerned in the spectra. A strong dielectric loss peak is observed in the clay with two layers of water molecules between the rigid clay platelets at temperatures between 125 and 215 K. This loss peak is neither observed in bulk ice, nor in the dry clay or clay with only one water layer between platelets. The relaxation time is shown to have an Arrhenius temperature dependence which does not extrapolate to the high-temperature behavior. This behavior is in accordance with the proposed so-called "fragile-strong" transition of supercooled bulk water, which implies a change in the temperature dependence of the relaxation time from a high-temperature non-Arrhenius to low-temperature Arrhenius behavior. However, this transition is experimentally elusive as it would occur in an inaccessible (due to the ease of crystallization) temperature range of the supercooled bulk water.