Automatic modulation classification (AMC) is a key step of signal demodulation that determines whether the receiver can correctly receive the transmitted signal without prior knowledge of the modulation type. Deep learning (DL) based AMC methods have been proven to achieve excellent performances. However, these DL-based methods rely heavily on expert experience to design neural network structures. These hand-designed networks have fixed architectures and lack flexibility, which often leads to insufficient model generalization. Neural architecture search (NAS) is a vital direction for automatic machine learning (AutoML), which can solve the shortcomings of hand-designed network architectures. In this paper, according to the specific modulation classification task, we propose a lightweight progressive differentiable architecture search-based AMC (PDARTS-AMC) method to search for a very lightweight network with great performance. In addition, the optimal architecture searched on dataset simulated by MATLAB is transferred to the RadioML2016.10B task, to verify the robustness and generalization of the proposed method. Experimental results show that the proposed PDARTS-AMC method both improves the classification accuracy and reduces the computational cost when compared with existing classical AMC methods.