We have detected the appearance of individual quantized vortex lines in liquid He II in narrow-diameter (~1 mm) rotating cylindrical vessels which were slowly accelerated from rest. The angular velocities where the lines appear are qualitatively consistent with theory based upon minimization of the free energy. However, we find that the detailed behavior of the He II is quite history dependent and cannot be uniquely determined from existing theory. Our detection technique involves charging the vortex lines for a fixed length of time. A subsequent measurement of the amount of charge trapped provides an indication of the number of vortex lines present. In order to detect the small amount of charge trapped on a single line, we have extracted the trapped electrons through the meniscus and have developed a proportional counter to detect them in the vapor.