Clouds of low energy electrons in the vacuum beam pipes of accelerators of positively charged particle beams present a serious limitation for operation of these machines at high currents. Because of the size of these accelerators, it is difficult to probe the low energy electron clouds over substantial lengths of the beam pipe. We have developed a novel technique to directly measure the electron cloud density via the phase shift induced in a TE wave that is independently excited and transmitted over a section of the accelerator. We infer the absolute phase shift with relatively high accuracy from the phase modulation of the transmission due to the modulation of the electron cloud density from a gap in the positively charged beam. We have used this technique for the first time to measure the average electron cloud density over a 50 m straight section in the positron ring of the PEP-II collider at the Stanford Linear Accelerator Center. We have also measured the variation of the density by using low field solenoid magnets to control the electrons. Show Partial Abstract