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This paper introduces a parameter adaptation-based control law for a class of nonlinear, control-affine, safety-critical systems subject to additive, parameter-affine model uncertainty. It is shown that the uncertainty is learned in fixed-time, i.e., within a finite time independent of the initial parameter estimates, without requiring persistence of excitation or rank conditions on the associated regressor matrix. A monotonically decreasing bound on the error of the estimated uncertainty is derived and used in the proposed control barrier function-based control law for guaranteed safety of the system trajectories. It is then proven that the parameter adaptation law is robust to bounded measurement noise and non-parametric perturbations to the system in that the estimated uncertainty still converges to a known neighborhood of the true uncertainty in fixed-time. The advantages of the approach are highlighted in a comparative numerical study, and the method is validated on a quadrotor trajectory-tracking problem subject to safety requirements and an unknown wind field.