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We compute the universal deformation ring of an odd Galois two dimensionalrepresentation of Gal$(M/Q)$ with an upper triangular image, where $M$ is themaximal abelian pro-$p$-extension of $F_{\infty}$ unramified outside a finiteset of places S, $F_{\infty}$ being a free pro-$p$-extension of a subextension$F$ of the field $K$ fixed by the kernel of the representation. We establish alink between the latter universal deformation ring and the universaldeformation ring of the representation of Gal$(K_S/Q)$, where $K_S$ is themaximal pro-$p$-extension of $K$ unramified outside $S$. We then give someexamples.  This paper was accepted for publication in the Mathematical Proceedings ofthe Cambridge philosophical society (May 99)

Computation of a universal deformation ring in the Borel case