Using video microscopy techniques, we have measured the position distribution of polystyrene spheres [P(z)] suspended in deionized water and confined between two glass plates. The sphere-plate interaction potential [U(z)] was deduced using equilibrium statistical mechanics : P(z) = C exp[-U(z)/k(B)T]. Measurements are presented for five different values of the plate spacing, and the results show that the rms displacement of the spheres, perpendicular to the plates, increases exponentially as the gap increases. The average position of the spheres is not always located at the midplane of the gap, indicating that there may be variations in the surface charge of the two glass plates. A comparison is made with the theory of Derjaguin, Landau, Verwey, and Overbeek (DLVO) for the interaction of colloidal particles with plane surfaces, and the experimental results are shown to be consistent with a harmonic approximation to the theory.