The Standard Model of particle physics assumes that the so-called fundamental constants are universal and unchanging. Absorption lines arising in molecular clouds along quasar sightlines offer a precise test for variations in the proton-to-electron mass ratio, mu, over cosmological time and distance scales. The inversion transitions of ammonia are particularly sensitive to mu as compared to molecular rotational transitions. Comparing the available ammonia spectra observed toward the quasar B0218+357 with new, high-quality rotational spectra, we present the first detailed measurement of mu with this technique, limiting relative deviations from the laboratory value to |Deltamu/mu| < 1.8 x 10(-6) (95% confidence level) at approximately half the universe's current age-the strongest astrophysical constraint to date. Higher-quality ammonia observations will reduce both the statistical and systematic uncertainties in these observations.