We have investigated the dynamics of confined water in a fully hydrated Na-vermiculite clay using the neutron spin-echo (NSE) technique. NSE measures the intermediate self-scattering function, i.e., the dynamics directly in the time domain. In the present experiments we performed measurements, in the time range 3-3000 ps and temperature ranging from 254 to 323 K, on the essentially two-dimensional water with a layer thickness corresponding to only two molecular layers. The data can be described by the Kohlrausch-Williams-Watts (KWW) stretched exponential function, probably indicating a broad distribution of relaxation times. The reason for the very stretched behavior of the intermediate self-scattering function I-S(Q,t), particularly in the supercooled regime, is most likely that the water molecules have widely different local environments. Some water molecules are strongly interacting with the clay surfaces or the intercalated Na+ ions, whereas the remaining molecules are interacting only with other water molecules. The average relaxation time for the dynamics observed in the experimental time window shows a pronounced non-Arrhenius temperature dependence, typical for the main relaxation of a "fragile" liquid, and a Q-dependence in reasonable agreement with the Gaussian jump-length distribution model for translational diffusion. The high temperature non-Arrhenius behavior of the relaxation time is in contrast to previous low temperature (125-215 K) dielectric results showing an Arrhenius temperature behavior, typical for a "strong" liquid. Although the NSE and dielectric data are not probing exactly the same water dynamics, the combined results suggests that our confined water undergoes a "fragile-strong" transition somewhere in the temperature range of 215-250 K.