A measurement apparatus has been constructed for the accurate determination of the densities of liquids and solids. A ball to a roller-bearing, made of stainless steel, is weighed in air and, hydrostatically, in a liquid of interest, using an electronic, single pan balance. Additionally, accurate mechanical measurements of the diameter, roundness and weight of the steel ball have been made. Several balls have been used, ranging in weight from 150 g to 1 kg. Measurements have been performed with pure water, and with kerosene. Extensive experiments have concentrated on the effects of surface tension on the hydrostatic weighing. Several choices of material for the suspension wire have been investigated. Some attempts to reduce the surface tension through application of an additive to the water are also reported. The density values obtained from the hydrostatic weighings are compared with the results of (i) the mechanical ball measurements; (ii) the tabulated density of pure water; (iii) measurements with areometers and pycnometers. The best results from hydrostatic weighings in pure water, with a wetted platinum suspension wire, gave a value for the density of steel of 7791.37 (±0.13) kg·m⁻³, compared with a mechanically-determined density of 7791.62 (±0.35) kg·m⁻³. Measurements of the density of kerosene with a variety of techniques were mostly in agreement. The best results being obtained with hydrostatic weighing in kerosene of density 766.21(8) kg·m⁻³ (at 19.4 °C).