The goal of our work was to select a neural network architecture that would give the best prediction of the Bitcoin exchange rate using historical data. Our work fits into the very important topic of predicting the value of the cryptocurrency exchange rate, and makes use of recent data which, as a result of the high Bitcoin exchange rate dynamics of the last year, differs significantly from those of previous years. We propose and test a number of neural network-based architectures and conduct a discussion of the results. Unlike previous state-of-the-art works, we conducted a comprehensive comparison of three different neural network-based models: MLP (multilayer perceptron), LSTM (long short-term memory) and CNN (convolutional neural network). We tested them for a wide range of parameters. The results we present are, to the best of our knowledge, the most up to date when it comes to the application of artificial intelligence methods for the prediction of cryptocurrency exchange rates. The best-performing architectures were used for a website that gives real-time predictions of the Bitcoin exchange rate. The website is available at http://stpbtc-ii.up.krakow.pl/. Source codes of our research are available to download in order to make our experiment reproducible.