We consider the first exit time of a Shiryaev-Roberts diffusion with constant positive drift from the interval $[0,A]$ where $A>0$. We show that the moment generating function (Laplace transform) of a suitably standardized version of the first exit time converges to that of the unit-mean exponential distribution as $A\to+\infty$. The proof is explicit in that the moment generating function of the first exit time is first expressed analytically and in a closed form, and then the desired limit as $A\to+\infty$ is evaluated directly. The result is of importance in the area of quickest change-point detection, and its discrete-time counterpart has been previously established - although in a different manner - by Pollak and Tartakovsky (2009).