We present DR.DECR (Dense Retrieval with Distillation-Enhanced Cross-Lingual Representation), a new cross-lingual information retrieval (CLIR) system trained using multi-stage knowledge distillation (KD). The teacher of DR.DECR relies on a highly effective but computationally expensive two-stage inference process consisting of query translation and monolingual IR, while the student, DR.DECR, executes a single CLIR step. We teach DR.DECR powerful multilingual representations as well as CLIR by optimizing two corresponding KD objectives. Learning useful representations of non-English text from an English-only retriever is accomplished through a cross-lingual token alignment algorithm that relies on the representation capabilities of the underlying multilingual encoders. In both in-domain and zero-shot out-of-domain evaluation, DR.DECR demonstrates far superior accuracy over direct fine-tuning with labeled CLIR data. It is also the best single-model retriever on the XOR-TyDi benchmark at the time of this writing.