  We prove an existence and uniqueness theorem for exact WKB solutions of
general singularly perturbed linear second-order ODEs in the complex domain.
These include the one-dimensional time-independent complex Schr\"odinger
equation. Notably, our results are valid both in the case of generic WKB
trajectories as well as closed WKB trajectories. We also explain in what sense
exact and formal WKB solutions form a basis. As a corollary of the proof, we
establish the Borel summability of formal WKB solutions for a large class of
problems, and derive an explicit formula for the Borel transform.
