Hyperspectral sensors acquire spectral responses from objects with a large number of narrow spectral bands. The large volume of data may be costly in terms of storage and computational requirements. In addition, hyperspectral data are often information-wise redundant. Band selection intends to overcome these limitations by selecting a small subset of spectral bands that provide more information or better performance for particular tasks. However, existing band selection techniques do not directly maximize the task-specific performance, but rather utilize hand-crafted metrics as a proxy to the final goal of performance improvement. In this paper, we propose a deep learning (DL) architecture composed of a constrained measurement learning network for band selection, followed by a classification network. The proposed joint DL architecture is trained in a data-driven manner to optimize the classification loss along band selection. In this way, the proposed network directly learns to select bands that enhance the classification performance. Our evaluation results with Indian Pines (IP) and the University of Pavia (UP) datasets show that the proposed constrained measurement learning-based band selection approach provides higher classification accuracy compared to the state-of-the-art supervised band selection methods for the same number of bands selected. The proposed method shows 89.08% and 97.78% overall accuracy scores for IP and UP respectively, being 1.34% and 2.19% higher than the second-best method.