The Automatic Dependent Surveillance-Broadcast (ADS-B) system is a key component of the Next Generation Air Transportation System (NextGen) that manages the increasingly congested airspace. It provides accurate aircraft localization and efficient air traffic management and also improves the safety of billions of current and future passengers. While the benefits of ADS-B are well known, the lack of basic security measures like encryption and authentication introduces various exploitable security vulnerabilities. One practical threat is the ADS-B spoofing attack that targets the ADS-B ground station, in which the ground-based or aircraft-based attacker manipulates the International Civil Aviation Organization (ICAO) address (a unique identifier for each aircraft) in the ADS-B messages to fake the appearance of non-existent aircraft or masquerade as a trusted aircraft. As a result, this attack can confuse the pilots or the air traffic control personnel and cause dangerous maneuvers. In this paper, we introduce SODA - a two-stage Deep Neural Network (DNN)-based spoofing detector for ADS-B that consists of a message classifier and an aircraft classifier. It allows a ground station to examine each incoming message based on the PHY-layer features (e.g., IQ samples and phases) and flag suspicious messages. Our experimental results show that SODA detects ground-based spoofing attacks with a probability of 99.34%, while having a very small false alarm rate (i.e., 0.43%). It outperforms other machine learning techniques such as XGBoost, Logistic Regression, and Support Vector Machine. It further identifies individual aircraft with an average F-score of 96.68% and an accuracy of 96.66%, with a significant improvement over the state-of-the-art detector.