We present the principle behind an experiment that will measure the time spent traversing the potential barrier during a tunneling event. This experiment is accomplished by measuring the macroscopic quantum tunneling rate out of the zero voltage state of a current-biased Josephson junction connected to a transmission line of adjustable length. The transmission line provides a frequency-dependent damping that reduces the tunneling rate. As the length of the line is increased from zero, the depression of the tunneling rate changes from that corresponding to the termination resistor to that corresponding to a resistor equal to the line impedance. This crossover occurs at a line delay time equal to the time spent traversing the barrier.