A detector has been constructed for measuring ion mobilities of gas mixtures at atmospheric pressure and room temperature. The detector consists of a standard triple GEM amplification region and a drift region where ions drift. A method has been developed to measure the ions arrival time at a cathode wire-grid by differentiating the recorded signals on this electrode. Simulations prove that this method is accurate and robust. The ion mobility in different gas mixtures is measured while applying different drift field values ranging from 200 V cm$^{-1}$ to 1100 V cm$^{-1}$. From an extrapolation of a Blancs law fit to measurements in Ar-CO$_2$ mixtures we find the reduced mobility of the drifting (cluster) ion species in pure argon to be $1.94pm0.01$ cm$^{2}$ V$^{-1}$ s$^{-1}$ and in pure carbon-dioxide to be $1.10pm0.01$ cm$^{2}$ V$^{-1}$ s$^{-1}$. Applying the same procedure to our measurements in Ne-CO$_2$ yields $4.06pm0.07$ cm$^{2}$ V$^{-1}$ s$^{-1}$ and $1.09pm0.01$ cm$^{2}$ V$^{- 1}$ s$^{-1}$ for the reduced mobilities in pure neon and carbon-dioxide, respectively. Admixtures of N$_2$ to Ne-CO$_2$ reduce somewhat the mobility. For the baseline gas mixture of the future ALICE Time Projection Chamber, Ne- CO$_2$-N$_{2}$ (90-10-5), the measured reduced mobility of the drifting ions is $2.92pm0.04$ cm$^{2}$ V$^{-1}$ s$^{-1}$. Ion mobilities are examined for different water content ranging from 70 ppm to about 2000 ppm in the gas using Ar-CO$_2$ (90-10) and Ne-CO$_2$ (90-10). A slight decrease of ion mobility is observed for the addition of several hundred ppm of water.
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