We report on the magnetic damping properties of polycrystalline $\mathrm{Fe}\text{\ensuremath{-}}\mathrm{V}$ alloy thin films that are deposited at room temperature. By varying the concentration of $\mathrm{V}$ in the alloy, the saturation magnetization can be adjusted from that of $\mathrm{Fe}$ to near zero. We show that exceptionally low values of the damping parameter can be maintained over the majority of this range, with a minimum damping at approximately 15%--20% $\mathrm{V}$ concentration. Such a minimum is qualitatively reproduced with ab initio calculations of the damping parameter, although at a concentration closer to 10% $\mathrm{V}$. The measured intrinsic damping has a minimum value of (1.53 \ifmmode\pm\else\textpm\fi{} 0.08) \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}3}$, which is approximately a factor of 3 higher than our calculated value of 0.48 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}3}$. From first-principles theory, we outline the factors that are mainly responsible for the trend of the damping parameter in these alloys. In particular, the band structure and resulting damping mechanism is shown to change at $\mathrm{V}$ concentrations greater than approximately 35% $\mathrm{V}$ content.
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