Beamsplitter cubes are widely used in white light interferometers to ensure that each interferometer path is balanced with equal amounts of glass. However, commercially available beamsplitter cubes can have alignment errors and size deviations that introduce path differences into the interferometer. A technique for constructing a beamsplitter cube that has extremely well balanced paths is described. Using a tunable dye laser with rhodimine 6G, we have readily measured path differences less than 0.5 μm. By monitoring the beamsplitter path difference with the dye laser, we can adjust the path difference during assembly, thereby sliding the prisms into proper alignment on a layer of UV-cured cement.