  Given a doubling measure $\mu$ on $R^d$, it is a classical result of harmonic
analysis that Calderon-Zygmund operators which are bounded in $L^2(\mu)$ are
also of weak type (1,1). Recently it has been shown that the same result holds
if one substitutes the doubling condition on $\mu$ by a mild growth condition
on $\mu$. In this paper another proof of this result is given. The proof is
very close in spirit to the classical argument for doubling measures and it is
based on a new Calderon-Zygmund decomposition adapted to the non doubling
situation.
