Variations of the Kelvin equation [W. Thomson, Philos. Mag. 42, 448 (1871)] to describe the freezing point depression of water in capillaries exist in the literature. The differing equations, coupled with the uncertainty in input parameters, lead to various predictions. The difference between the predictions may become substantial when the capillary size decreases much below micron dimensions. An experiment was designed to investigate the predicted values using a customized directional solidification stage. The capillary freezing point depression for glass tubes with radii of 87 microm-3 microm was successfully measured. The image of the ice-water interface at equilibrium was also digitally captured and analyzed to examine the contact angle and the interface shape as well. Both are important for examining the hemispherical interface assumption that was exclusively used in the theoretical derivations. Finally, an equilibrium analysis of the thermodynamic system leads to a theoretical discussion of the problem. The effect of the temperature gradient on the interface shape is addressed, and an engineering criterion for the critical temperature gradient above which the effect must be considered for the interface shape calculation is derived.