While much of synthetic biology was founded on the creation of reusable, standardized parts, there is now a growing interest in synthetic networks which can compute unique, specially-designed functions in order to recognize patterns or classify cells in-vivo. While artificial neural networks (ANNs) have long provided a mature mathematical framework to address this problem in-silico, their implementation becomes much more challenging in living systems. In this work, we propose a Biomolecular Neural Network (BNN), a dynamical chemical reaction network which faithfully implements ANN computations and which is unconditionally stable with respect to its parameters when composed into deeper networks. Our implementation emphasizes the usefulness of molecular sequestration for achieving negative weight values and a nonlinear activation function in its elemental unit, a biomolecular perceptron. We then discuss the application of BNNs to linear and nonlinear classification tasks, and draw analogies to other major concepts in modern machine learning research.