The European Southern Observatory's planned Astronomical Weather Station for the Very Large Telescope which is currently under construction at Cerro Paranal in Chile includes (i) advance temperature prediction, which would permit air conditioning in the telescope enclosure to be preset as a function of the next night's expected temperature; and (ii) prediction of seeing, a few hours in advance, to allow flexible scheduling of the most appropriate instrumentation. Extensive data, collected since 1985, are being used to appraise various methodologies. A recurrent neural network is described, which uses arbitrary time-delayed connections to capture the dynamic of time series. This endows the model with a memory of its previous states. The resulting network is time- and space-recurrent, and generalizes most recurrent architectures. The performance of this network is discussed. The results are compared with the k-nearest neighbors method.