The phase change caused by excitation of surface plasmons (SP) in a Kretschmann configuration was measured by a rotating analyzer scheme. The set-up was used to determinate the optical constants at 632.8 nm of an indium tin oxide (ITO) film with a thickness of 20 nm on a gold film. The numerical values n=0.182, k=3.427 for the An film, and n=1.91, k=0.01 for ITO were obtained by a best-fit procedure to experimental data for several gold thicknesses. Poynting's vector calculations show that the energy of the plasmon field is redistributed after deposition of the ITO layer. This causes a shift in both the surface plasmon resonance minimum and the turning point of the phase curve. From model calculations it is argued that in thin-film cases SP phase measurements give more precise values for the optical constants than conventional ellipsometry and SP amplitude methods.