  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. $\sigma^{-1} \in L^1\left (\frac{1}{\lambda} {\bf
d}\lambda\right )$. 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 $\text{o}(|x-x_0|)$. 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 the
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.
