O-RAN is a new, open, adaptive, and intelligent RAN architecture. Motivated by the success of artificial intelligence in other domains, O-RAN strives to leverage machine learning (ML) to automatically and efficiently manage network resources in diverse use cases such as traffic steering, quality of experience prediction, and anomaly detection. Unfortunately, it has been shown that ML-based systems are vulnerable to an attack technique referred to as adversarial machine learning (AML). This special kind of attack has already been demonstrated in recent studies and in multiple domains. In this paper, we present a systematic AML threat analysis for O-RAN. We start by reviewing relevant ML use cases and analyzing the different ML workflow deployment scenarios in O-RAN. Then, we define the threat model, identifying potential adversaries, enumerating their adversarial capabilities, and analyzing their main goals. Next, we explore the various AML threats associated with O-RAN and review a large number of attacks that can be performed to realize these threats and demonstrate an AML attack on a traffic steering model. In addition, we analyze and propose various AML countermeasures for mitigating the identified threats. Finally, based on the identified AML threats and countermeasures, we present a methodology and a tool for performing risk assessment for AML attacks for a specific ML use case in O-RAN.