The ambient calculus is a process calculus for describing mobile computation. We develop a theory of Morris-style contextual equivalence for proving properties of mobile ambients. We prove a context lemma that allows derivation of contextual equivalences by considering contexts of a particular limited form, rather than all arbitrary contexts. We give an activity lemma that characterises the possible interactions between a process and a context. We prove several examples of contextual equivalence. The proofs depend on characterising reductions in the ambient calculus in terms of a labelled transition system.