We show that a continuous, odd, regular (non-exceptional), ordinary, irreducible, two-dimensional, $l$-adic representation of the absolute Galois group of the rational numbers is modular over some totally real field. We deduce that it occurs in the $l$-adic cohomology of some variety over the rationals and that its $L$-function has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.

AMS 2000 Mathematics subject classification: Primary 11F80. Secondary 11G40