Speech is converted to digital signals using speech coding for efficient transmission. However, this often lowers the quality and bandwidth of speech. This paper explores the application of convolutional neural networks for Artificial Bandwidth Expansion (ABE) and speech enhancement on coded speech, particularly Adaptive Multi-Rate (AMR) used in 2G cellular phone calls. In this paper, we introduce AMRConvNet: a convolutional neural network that performs ABE and speech enhancement on speech encoded with AMR. The model operates directly on the time-domain for both input and output speech but optimizes using combined time-domain reconstruction loss and frequency-domain perceptual loss. AMRConvNet resulted in an average improvement of 0.425 Mean Opinion Score - Listening Quality Objective (MOS-LQO) points for AMR bitrate of 4.75k, and 0.073 MOS-LQO points for AMR bitrate of 12.2k. AMRConvNet also showed robustness in AMR bitrate inputs. Finally, an ablation test showed that our combined time-domain and frequency-domain loss leads to slightly higher MOS-LQO and faster training convergence than using either loss alone.