With recent advances in smartphones and wearable sensors, Body Sensor Networks (BSNs) have been proposed for use in continuous, remote electrocardiogram (ECG) monitoring. In such systems, sampling the ECG at clinically recommended rates (250 Hz) and wireless transmission of the collected data incurs high energy consumption at the energy-constrained body sensor. The large volume of collected data also makes data storage at the sensor infeasible. Thus, there is a need for reducing the energy consumption and data size at the sensor, while maintaining the ECG quality required for diagnosis. In this paper, we propose GeM-REM, a resource-efficient ECG monitoring method for BSNs. GeM-REM uses a generative ECG model at the base station and its lightweight version at the sensor. The sensor transmits data only when the sensed ECG deviates from model-based values, thus saving transmission energy. Further, the model parameters are continually updated based on the sensed ECG. The proposed approach enables storage of ECG data in terms of model parameters rather than data samples, which reduces the required storage space. Implementation on a sensor platform and evaluation using real ECG data from MIT-BIH dataset shows transmission energy and data storage reduction ratios of 42.1:1 and 37.3:1 respectively, which are better than state of the art ECG data compression schemes.