<p> Searches for violations of discrete symmetries can be sensitive probes of physics beyond the Standard Model. Many models, such as supersymmetric theories, introduce new particles at higher masses that include new <i> CP</i>-violating phases which are thought to be of order unity. Such phases could generate measurable permanant electric dipole moments (EDMs) of particles. The ACME collaboration has measured the electron's EDM to be consistent with zero with an order of magnitude improvement in precision compared to the previous best precision (J. Baron et al., ACME collaboration, <i> Science</i> <b>343</b> (2014), 269-272) with a spin precession measurement performed in the <i>H</i> state of a beam of thorium monoxide (ThO). This limit constrains time-reversal violating physics for particles with masses well into the TeV scale. In this thesis I discuss the details of this measurement with an emphasis on the data analysis, search for systematic errors, and systematic error models that contributed to this result. I also discuss implemented and planned upgrades to the experimental apparatus intended to both improve the statistical sensitivity and reduce its susceptibility to systematic errors. At this time, the upgraded apparatus has been demonstrated to have a statistical sensitivity to the electron EDM that is more than a factor of 10x more precise than our previous measurement. </p><p>