We demonstrate an atom-based amplitude-modulation (AM) receiver for digital communication with a weak continuous frequency carrier using a Rydberg AC Stark effect in a vapor cell and achieve the operating carrier frequency continuously from 0.1 GHz to 5 GHz at a single Rydberg state. A strong local oscillator (LO) field ELO acts as a gain to shift the Rydberg level to a high sensitivity region, and a weak carrier field ECarr keeps the same frequency with the LO field. The digital baseband signals are encoded onto the ECarr using the amplitude modulation technique with the different modulation frequency. The response of Rydberg atom to the baseband signal is probed via a Rydberg electromagnetically induced transparency (EIT). The measured instantaneous bandwidth of the system is about 230 kHz. To demonstrate the performance of our system for an actual communication, we consider a color image as an example, the received image displays that the bit error rate (BER) is less than 5% when the maximum data transfer rate is about 238 kbps. Meanwhile, our system shows the weak carrier field of ECarr ≥ 13.52 μV/cm can be used for the practical communication with BER less than 5%. Our works break the limitation that EIT-AT based atomic receivers only operate at the near resonant frequencies of the Rydberg transitions, making this emerging of quantum technology close to the practical application with high sensitivity and broad bandwidth.