Due to their high degree of parallelism, fast processing speeds and low power consumption, analog optical functional elements offer interesting routes for realizing neuromorphic computer hardware. For instance, convolutional neural networks lend themselves to analog optical implementations by exploiting the Fourier-transform characteristics of suitable designed optical setups. However, the efficient implementation of optical nonlinearities for such neural networks still represents challenges. In this work, we report on the realization and characterization of a three-layer optical convolutional neural network where the linear part is based on a 4f-imaging system and the optical nonlinearity is realized via the absorption profile of a cesium atomic vapor cell. This system classifies the handwritten digital dataset MNIST with 83.96% accuracy, which agrees well with corresponding simulations. Our results thus demonstrate the viability of utilizing atomic nonlinearities in neural network architectures with low power consumption.