  We prove that there exists essentially one {\it minimal} differential algebra
of distributions $\A$, satisfying all the properties stated in the Schwartz
impossibility result [L. Schwartz, Sur l'impossibilit\'e de la multiplication
des distributions, 1954], and such that $\C_p^{\infty} \subseteq \A \subseteq
\DO' $ (where $\C_p^{\infty}$ is the set of piecewise smooth functions and
$\DO'$ is the set of Schwartz distributions over $\RE$). This algebra is
endowed with a multiplicative product of distributions, which is a
generalization of the product defined in [N.C.Dias, J.N.Prata, A multiplicative
product of distributions and a class of ordinary differential equations with
distributional coefficients, 2009]. If the algebra is not minimal, but
satisfies the previous conditions, is closed under anti-differentiation and the
dual product by smooth functions, and the distributional product is continuous
at zero then it is necessarily an extension of $\A$.
