An in situ structural investigation of the underpotential deposition (UPD) of copper on an iodine-covered platinum surface (Pt/C layered synthetic microstructure with Pt as the outermost layer) has been carried out using surface extended X-ray absorption fine structure (SEXAFS) and its polarization dependence. The effects of rinsing the electrode with pure supporting electrolyte have also been investigated. At an applied potential of +0.20 V (corresponding to half a monolayer of electrodeposited opper) there are two in-plane Cu-Cu distances : 2.56 +/- 0.05 Angstrom and 4.21 +/- 0.05 A. This suggests the coexistence of a close-packed phase with a more open phase. It is likely that the open phase is stabilized by repulsive interactions between partially charged copper atoms, as well as by the presence of the strongly adsorbed iodine layer. At +0.10 V (corresponding to a full monolayer of electrodeposited copper) there is only one Cu-Cu distance of 2.56 +/- 0.05 Angstrom. Combining in-plane (parallel to the electrode surface) and out-of-plane (perpendicular to the electrode surface) SEXAFS measurements, we found that at full monolayer coverage, the copper UPD layer is incommensurate with respect to the platinum surface. The iodine ad-layer rides the eletrodeposited copper forming a 3 x 3 unit cell. This interfacial structure is not altered upon rinsing with pure supporting electrolyte.