The need to recover high-dimensional signals from their noisy low-resolution quantized measurements is widely encountered in communications and sensing. In this paper, we focus on the extreme case of one-bit quantizers, and propose a deep detector network, called LoRD-Net, for signal recovering from one-bit measurements. Our approach relies on a model-aware data-driven architecture, based on a deep unfolding of first-order optimization iterations. LoRD-Net has a task-based architecture dedicated to recovering the underlying signal of interest from the one-bit noisy measurements without requiring prior knowledge of the channel matrix through which the one-bit measurements are obtained. The proposed deep detector has much fewer parameters compared to black-box deep networks due to the incorporation of domain-knowledge in the design of its architecture, allowing it to operate in a data-driven fashion while benefiting from the flexibility, versatility, and reliability of model-based optimization methods. We numerically evaluate the proposed receiver architecture for one-bit signal recovery in wireless communications and demonstrate that the proposed hybrid methodology outperforms both data-driven and model-based state-of-the-art methods, while utilizing small datasets, on the order of merely 500 samples, for training.