Automatically crafting test scenarios for REST APIs helps deliver more reliable and trustworthy web-oriented systems. However, current black-box testing approaches rely heavily on the information available in the API's formal documentation, i.e., the OpenAPI Specification (OAS for short). While useful, the OAS mostly covers syntactic aspects of the API (e.g., producer-consumer relations between operations, input value properties, and additional constraints in natural language), and it lacks a deeper understanding of the API business logic. Missing semantics include implicit ordering (logic dependency) between operations and implicit input-value constraints. These limitations hinder the ability of black-box testing tools to generate truly effective test cases automatically. This paper introduces DeepREST, a novel black-box approach for automatically testing REST APIs. It leverages deep reinforcement learning to uncover implicit API constraints, that is, constraints hidden from API documentation. Curiosity-driven learning guides an agent in the exploration of the API and learns an effective order to test its operations. This helps identify which operations to test first to take the API in a testable state and avoid failing API interactions later. At the same time, experience gained on successful API interactions is leveraged to drive accurate input data generation (i.e., what parameters to use and how to pick their values). Additionally, DeepREST alternates exploration with exploitation by mutating successful API interactions to improve test coverage and collect further experience. Our empirical validation suggests that the proposed approach is very effective in achieving high test coverage and fault detection and superior to a state-of-the-art baseline.