This note investigates the control of stochastic nonlinear systems with parametric uncertainty. The class of systems considered are single-input-single-output and in strict-feedback form, with the performance measured with respect to a risk-sensitive cost criterion. The uncertainty in the system description is assumed to be linearly parameterized, where the unmeasured parameters are generated by stochastic differential equations. By employing the backstepping design technique on the estimates of the unmeasured states, provided by a simple state estimator, an output-feedback adaptive controller is constructed which maintains an arbitrarily small average value for the risk-sensitive cost. The controller designed achieves boundedness in probability for all closed-loop signals and, under certain conditions, the tracking error converges to zero almost surely.