The uniformity of the helium-to-hydrogen (He-to-H) abundance ratio in the X-ray emitting intracluster medium (ICM) is one of the commonly adopted assumptions in X-ray analyses of galaxy clusters and cosmological constraints derived from these measurements. In this paper, we investigate the effect of He sedimentation on X-ray measurements of galaxy clusters in order to assess this assumption and associated systematic uncertainties. By solving a set of flow equations for a H-He plasma, we show that the He-to-H mass ratio is significantly enhanced in the inner regions of clusters. The effect of He sedimentation, if not accounted for, introduces systematic biases in observable properties of clusters derived using X-ray observations. We show that these biases also introduce an apparent evolution in the observed gas mass fractions of X-ray luminous, dynamically relaxed clusters and hence biases in observational constraints on the dark energy equation of state parameter, w, derived from the cluster distance-redshift relation. The Hubble parameter derived from the combination of X-ray and Sunyaev-Zel'dovich effect measurements is affected by the He sedimentation process as well. Future measurements aiming to constrain w or H ₀ to better than 10% may need to take into account the effect of He sedimentation. We propose that the evolution of gas mass fraction in the inner regions of clusters should provide unique observational diagnostics of the He sedimentation process.