A concise derivation of a new multiplicative product of Schwartz distributions is presented. The new product $\star$ is defined in the vector space ${\cal A}$ of piecewise smooth functions on $\bkR$ and all their (distributional) derivatives; it is associative, satisfies the Leibnitz rule and reproduces the usual product of functions for regular distributions. The algebra $({\cal A},+,\star)$ yields a sufficiently general setting to address some interesting problems. As an application we consider the problem of deriving a global formulation for quantum confined systems., Comment: 5 pages, no figures. Presented by N.C. Dias at the 3rd Baltic-Nordic Workshop "Algebra, Geometry, and Mathematical Physics", Goteborg, Sweden, October 11-13, 2007