Direct model reference adaptive control of linear uncertain dynamic systems with concurrent learning update laws in the presence of noisy measurements is investigated. Previous results have shown that in the absence of noise, concurrent learning adaptive controllers, which use specifically selected and online recorded data concurrently with instantaneous data for adaptation, can guarantee exponential tracking and weight error convergence without requiring persistency of excitation. Here the robustness of a concurrent learning adaptive controller to noisy measurements is established. A Lyapunov framework is employed to show boundedness of tracking and weight error dynamics in a compact ball around the origin in the presence of bounded measurement noise. Furthermore, it is shown that when concurrent learning is used, additional modification terms such as - or e-modification are not required for guaranteeing robustness in the presence of noise. Numerical simulations and experimental results demonstrate improved performance.