T2K is a long baseline neutrino oscillation experiment which uses a beam of muon neutrinos to study muon neutrino disappearance and electron neutrino appearance. The measurement of v_e appearance is sensitive to CP violation, and reducing the systematic uncertainties is critical to enabling experimental determination of whether there is CP violation in the lepton sector. This thesis describes the first measurement of the v_e CC inclusive cross-section on carbon at energies relevant to T2K and other long baseline neutrino oscillation experiments. The T2K near detector, ND280, is used to select a sample of v_e events, and a Bayesian unfolding technique is used to extract differential cross-sections as a function of electron momentum, electron angle and Q^2. The total flux-averaged cross-section is also extracted, and is found to be 1.11 ± 0.20 x 10^{-38} cm^2/nucleon, which agrees well with both the NEUT neutrino interaction generator prediction of 1.23 x 10^{-38} cm^2/nucleon and the GENIE prediction of 1.08 x 10^{-38} cm^2/nucleon. A restricted phase-space analysis is also performed, in which only events where the ejected electron has momentum greater than 550 MeV/c and cos(θ_e) > 0.72 are selected. In this case, the flux-averaged cross-section is found to be 6.54 ± 1.22 x 10^{-39} cm^2/nucleon, again in good agreement with both the NEUT prediction of 7.38 x 10^{-39} cm^2/nucleon and the GENIE prediction of 6.41 x 10^{-39} cm^2/nucleon.