The virtual exchange of new, heavy particles can lead to measurable effects in ordinary matter. Such underlying physics can induce a permanent electric dipole moment (EDM) of the electron, which alters structure of neutral atoms or molecules in a distinctive and predictable way. A wide variety of extensions to the Standard Model predict a value for the electron EDM within reach of the next generation of experiments. I will describe the experiment which gives the present upper limit on the electron EDM, and progress towards a new method which should yield an improvement in sensitivity by up to four orders of magnitude.