We have detected narrow HI 21cm and CI absorption at $z sim 1.4 - 1.6$ towards Q0458$-$020 and Q2337$-$011, and use these lines to test for possible changes in the fine structure constant $alpha$, the proton-electron mass ratio $mu$, and the proton gyromagnetic ratio $g_p$. A comparison between the HI 21cm and CI line redshifts yields $Delta X/X = [+6.8 pm 1.0] times 10^{-6}$ over $0 < <z> le 1.46$, where $X = g_p alpha^2/mu$, and the errors are purely statistical, from the gaussian fits. The simple line profiles and the high sensitivity of the spectra imply that statistical errors in this comparison are an order of magnitude lower than in previous studies. Further, the CI lines arise in cold neutral gas that also gives rise to HI 21cm absorption, and both background quasars are core-dominated, reducing the likelihood of systematic errors due to local velocity offsets between the hyperfine and resonance lines. The dominant source of systematic error lies in the absolute wavelength calibration of the optical spectra, which appears uncertain to $sim 2$ km/s, yielding a maximum error in $Delta X/X$ of $sim 6.7 times 10^{-6}$. Including this, we obtain $Delta X/X = [+6.8 pm 1.0 (statistical) pm 6.7 (max. systematic)] times 10^{-6}$ over $0 < <z> le 1.46$. Using literature constraints on $Delta mu/mu$, this is inconsistent with claims of a smaller value of $alpha$ from the many-multiplet method, unless fractional changes in $g_p$ are larger than those in $alpha$ and $mu$.
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