Accurate potential measurements in electrophysiological experiments require correction for liquid junction potentials (LJPs), and, in patch-clamping especially, these can often be ~5-10 mV or more. They can be either calculated, if ion mobilities are known, or measured directly. We describe an optimised system to directly measure LJPs with a patch-clamp amplifier, using as a reference electrode, a freshly-cut 3 M KCl-agar salt-bridge (in polyethylene tubing) with its tip cut off by at least 5 mm during solution changes to eliminate its solution-history-dependent effects. We quantify such history-dependent effects and complement this with a de-novo theoretical analysis of salt diffusion to and from the salt-bridge. Our analysis and experimental results validate the optimised methodology for measuring LJPs, and the use of the Henderson equation for accurately calculating them. The use of this equation is also assessed and generally validated in the light of rigorous Nernst-Planck-Poisson and other numerical simulations and analytical studies of LJPs over recent decades. Digitizing, recording and amplifying the measured potentials increases their accuracy. The measured potentials still need correction for small, well-defined calculable, shifts in LJPs at the 3 M KCl-agar reference. Using this technique, we have measured changes in LJPs for diluted solutions of NaCl, LiCl, KCl, CsCl and NaF, obtaining excellent agreement within ±0.1 mV of predicted values, calculated using ion activities. Our de novo LJP measurements of biionic combinations of the above undiluted salts, and NaI and NaF (with halide anions I- and F-), generally also gave excellent agreement with predicted values.