We propose a new one-parameter family of Landau gauges for Yang-Mills theories which can be formulated by means of functional integral methods and are thus well suited for analytic calculations, but which are free of Gribov ambiguities and avoid the Neuberger zero problem of the standard Faddeev-Popov construction. The resulting gauge-fixed theory is perturbatively renormalizable in four dimensions and, for what concerns the calculation of ghost and gauge field correlators, it reduces to a massive extension of the Faddeev-Popov action. We study the renormalization group flow of this theory at one-loop and show that it has no Landau pole in the infrared for some - including physically relevant - range of values of the renormalized parameters.