Most existing semantic communication systems employ analog modulation, which is incompatible with modern digital communication systems. Although several digital transmission approaches have been proposed to address this issue, an end-to-end bit-level method that is compatible with arbitrary modulation formats, robust to channel noise, and free from quantization errors remains lacking. To this end, we propose BitSemCom, a novel bit-level semantic communication framework that realizes true joint source-channel coding (JSCC) at the bit level. Specifically, we introduce a modular learnable bit mapper that establishes a probabilistic mapping between continuous semantic features and discrete bits, utilizing the Gumbel-Softmax trick to enable differentiable bit generation. Simulation results on image transmission demonstrate that BitSemCom achieves both competitive performance and superior robustness compared to traditional separate source-channel coding (SSCC) schemes, and outperforms deep learning based JSCC with uniform 1-bit quantization, validating the effectiveness of the learnable bit mapper. Despite these improvements, the bit mapper adds only 0.42% parameters and 0.09% computational complexity, making BitSemCom a lightweight and practical solution for real-world semantic communication.