Automatic modulation classification (AMC) is an promising technology for non-cooperative communication systems in both military and civilian scenarios. Recently, deep learning (DL) based AMC methods have been proposed with outstanding performances. However, both high computing cost and large model sizes are the biggest hinders for deployment of the conventional DL based methods, particularly in the application of internet-of-things (IoT) networks and unmanned aerial vehicle (UAV)-aided systems. In this correspondence, a novel DL based lightweight AMC (LightAMC) method is proposed with smaller model sizes and faster computational speed. We first introduce a scaling factor for each neuron in convolutional neural network (CNN) and enforce scaling factors sparsity via compressive sensing. It can give an assist to screen out redundant neurons and then these neurons are pruned. Experimental results show that the proposed LightAMC method can effectively reduce model sizes and accelerate computation with the slight performance loss.