  The singularly perturbed Riccati equation is the first-order nonlinear ODE
$\hbar \partial_x f = af^2 + bf + c$ in the complex domain where $\hbar$ is a
small complex parameter. We prove an existence and uniqueness theorem for exact
solutions with prescribed asymptotics as $\hbar \to 0$ in a halfplane. These
exact solutions are constructed using the Borel-Laplace method; i.e., they are
Borel summations of the formal divergent $\hbar$-power series solutions. As an
application, we prove existence and uniqueness of exact WKB solutions for the
complex one-dimensional Schr\"odinger equation with a rational potential.
