We recently set a new limit on the electric dipole moment of the electron (eEDM) (J Baron et al and ACME collaboration 2014 Science 343 269-272), which represented an order-of-magnitude improvement on the previous limit and placed more stringent constraints on many charge-parityviolating extensions to the standard model. In this paper we discuss the measurement in detail. The experimental method and associated apparatus are described, together with the techniques used to isolate the eEDM signal. In particular, we detail the way experimental switches were used to suppress effects that can mimic the signal of interest. The methods used to search for systematic errors, and models explaining observed systematic errors, are also described. We briefly discuss possible improvements to the experiment. 9 Note that the limit we report here uses an updated value for = 78 eff GV cm −1 which is obtained by averaging the results from [29, 30]. 10 A detailed discussion of the sign convention for this Hamiltonian term is provided in section appendix A.3 1 states have very small magnetic moments [58] since the d 3 2 orbital valence electron serves to nearly cancel the magnetic moment of the s 1 2 orbital. The actual magnetic moment of H deviates from zero primarily because of mixing with other states [59]. We express ThO molecule states using the basis Wñ |Y J M , , , , where Y is the electronic state, J is the total angular momentum, M is the projection of J onto the laboratoryẑ-axis, and Ω 1 (V cm −1 ) −1 [67]; this permits full (>99%) polarisation of the state in small applied electric fields, 10 V cm −1 , allowing us to take full advantage of the huge eff in ThO. The Ω-doublet structure is also useful in rejecting systematic errors since it allows for spectroscopic reversal of µ -n eff by addressing different states without reversing the applied electric field [68]. This is discussed in greater detail in section 5.4.The H state in ThO is metastable with a lifetime »1.8 ms [69], limiting our measurement time to t » 1 ms. We note that this is comparable to previous beam-based eEDM measurements where the atomic/molecular states used had significantly longer lifetimes [20,69,70]. 1 (V cm −1 ) −1 (black arrow/lines) is the expectation value of the molecular electric dipole moment in these states [60]. Additionally, a magnetic field causes a Zeeman shift m »-Mg z 1 B , with m p » -ǵ 2 6kHz 1 B G −1 (red arrow/lines) [59, 64]. A nonzero eEDM would result in an additional energy shift »-M d e eff (blue arrow/lines) where = -1 (+1) when the applied field is (is not) reversed. The orientation of eff (green arrows), the spin of the electron in the σ orbital (black arrow next to molecule), the external electric field , and the external magnetic field are shown relative to the laboratoryẑ direction which is oriented upwards on the page. Diagram not to scale. 11 Throughout the paper, we give numerical values of energies (with = 1) in terms of angular frequencies by using the notation p´f 2 , where f ...