We present a novel machine-learning approach for detecting faint point sources in high-contrast adaptive optics (AO) imaging data sets. The most widely used algorithms for primary subtraction aim to decouple bright stellar speckle noise from planetary signatures by subtracting an approximation of the temporally evolving stellar noise from each frame in an imaging sequence. Our approach aims to improve the stellar noise approximation and increase the planet detection sensitivity by leveraging deep learning in a novel direct imaging post-processing algorithm. We show that a convolutional autoencoder neural network, trained on an extensive reference library of real imaging sequences, accurately reconstructs the stellar speckle noise at the location of a potential planet signal. This tool is used in a post-processing algorithm we call Direct Exoplanet Detection with Convolutional Image Reconstruction, or ConStruct. The reliability and sensitivity of ConStruct are assessed using real Keck/NIRC2 angular differential imaging data sets. Of the 30 unique point sources we examine, ConStruct yields a higher signal-to-noise ratio than traditional principal component analysis-based processing for 67% of the cases and improves the relative contrast by up to a factor of 2.6. This work demonstrates the value and potential of deep learning to take advantage of a diverse reference library of point-spread function realizations to improve direct imaging post-processing. ConStruct and its future improvements may be particularly useful as tools for post-processing high-contrast images from JWST and extreme AO instruments, both for the current generation and those being designed for the upcoming 30 m class telescopes.