We prove that any solution of a degenerate elliptic PDE is of class C1 provided the inverse of the equation's degeneracy law satisfies an integrability criterium, viz. σ−1∈L1(1λdλ). The proof is based upon the construction of a sequence of converging tangent hyperplanes that approximate u(x), near x0, by an error of order o(|x−x0|). Explicit control of such hyperplanes is carried over through the construction, yielding universal estimates upon the C1–regularity of solutions. Among the main new ingredients required in the proof, we develop an alternative recursive algorithm for renormalization of approximating solutions. This new method is based on a technique tailored to prevent the sequence of degeneracy laws constructed through the process from being, itself, degenerate.