Automatic Modulation Classification (AMC) plays a vital role in time series analysis, such as signal classification and identification within wireless communications. Deep learning-based AMC models have demonstrated significant potential in this domain. However, current AMC models inadequately consider the disparities in handling signals under conditions of low and high Signal-to-Noise Ratio (SNR), resulting in an unevenness in their performance. In this study, we propose MoE-AMC, a novel Mixture-of-Experts (MoE) based model specifically crafted to address AMC in a well-balanced manner across varying SNR conditions. Utilizing the MoE framework, MoE-AMC seamlessly combines the strengths of LSRM (a Transformer-based model) for handling low SNR signals and HSRM (a ResNet-based model) for high SNR signals. This integration empowers MoE-AMC to achieve leading performance in modulation classification, showcasing its efficacy in capturing distinctive signal features under diverse SNR scenarios. We conducted experiments using the RML2018.01a dataset, where MoE-AMC achieved an average classification accuracy of 71.76% across different SNR levels, surpassing the performance of previous SOTA models by nearly 10%. This study represents a pioneering application of MoE techniques in the realm of AMC, offering a promising avenue for elevating signal classification accuracy within wireless communication systems.