The electromagnetic form factors of the neutron and proton, G[E]n,p and G[M]n,p are of fundamental importance to our understanding of their internal structure, and provide useful constraints for models of nucleon structure. Until recently, measurements of the neutron electric form factor, Gn[E], were hampered by the model dependence of e-D scattering data, and from the small size of GnE relative to Gn[M]. New techniques using polarised electron beams offer improved methods of measurement where Gn[E] appears in the product Gn[E]Gn[M] as an interference term. This thesis comprises such a measurement, which was performed recently at the University of Mainz's MAMI-B electron accelerator facility, using the reaction e(D,e'n)p, where the recoil neutron transverse polarisation component P[x] gives a measure of Gn[E]. The thesis includes a study of the neutron polarimeter using a Monte-Carlo model which provides an estimate of the analysing power. The model was developed using the particle physics simulation package GEANT and the neutron detection efficiency program STANTON, requiring substantial modification to allow the study of the passage of neutrons through a complex detector geometry, and to simulate the asymmetric neutron distribution following p(n,np) scattering. The simulation results are in good agreement with those of an independent simulation ([21]). The polarimeter is calibrated by the use of a spin-precession technique, in which the longitudinal P[z] component of neutron polarisation is precessed by a magnetic field of integral ~1.5Tm by ninety degrees, and hence becomes transverse and measurable by the polarimeter. The P[z] component is essentially independent of Gn[E] and is calculable, allowing the polarimeter analysing power to be determined. Analysing powers of around 15% are found, in good agreement with the results of the Monte-Carlo simulation. A preliminary measurement of Gn[E] is made at Q2 = 9.4 fm-2 by measuring the transverse P[x] component.