In this paper we use Baker theory for giving an alternative proof of

Leopoldt's Conjecture for totally real extensions $\K$. This approach uses a
formulation of the Conjecture for relative extensions which can be proved by
Diophantine approximation and reduces the problem to the fact that $\rg{B}$,
the module of classes containing products of $p$ - units, is finite. The proof
of this fact is elementary, but requires class field theory. The methods used
here are a sharpening of the ones presented at the SANT meeting in G\"ottingen,
2008 and exposed in \cite{Mi1}, \cite{Mi2}.