Incalculable numbers of patients in hospitals as a result of COVID-19 made the screening of heart patients arduous. Patients who need regular heart monitoring were affected the most. Telecardiology is used for regular remote heart monitoring of such patients. However, the resultant huge electrocardiogram (ECG) data obtained through regular monitoring affects available storage space and transmission bandwidth. These signals can take less space if stored or sent in a compressed form. To recover them at the receiver end, they are decompressed. We have combined telecardiology with automatic ECG arrhythmia classification using CNN and proposed an algorithm named TELecardiology using a Deep Convolution Neural Network (TELDCNN). Discrete cosine transform (DCT), 16-bit quantization, and run length encoding (RLE) were used for compression, and a convolution neural network (CNN) was applied for classification. The database was formed by combining real-time signals (taken from a designed ECG device) with an online database from Physionet. Four kinds of databases were considered and classified. The attained compression ratio was 2.56, and the classification accuracies for compressed and decompressed databases were 0.966 and 0.990, respectively. Comparing the classification performance of compressed and decompressed databases shows that the decompressed signals can classify the arrhythmias more appropriately than their compressed-only form, although at the cost of increased computational time.