We prove that any solution of a degenerate elliptic PDE is of class C 1 provided the inverse of the equation's degeneracy law satisfies an integrability criterium, viz. σ− 1∈ L 1 (1 λ d λ). The proof is based upon the construction of a sequence of converging tangent hyperplanes that approximate u (x), near x 0, by an error of order o (| x− x 0|). Explicit control of such hyperplanes is carried over through the construction, yielding universal estimates upon the C 1–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.