AbstractWe prove Serre’s conjecture for the case of Galois representationsof Serre’s weight 2 and level 1. We do this by combining the poten-tial modularity results of Taylor and lowering the level for Hilbertmodular forms with a Galois descent argument, properties of univer-sal deformation rings, and the non-existence of p -adic Barsotti-Tateconductor 1 Galois representations proved in [6]. 1. Introduction In this article we prove the non-existence of odd, two-dimensional, irre-ducible representations of the absolute Galois group of Q with values in afinite field of odd characteristic p , in the case of Serre’s weight 2 and level(conductor) 1.Equivalently, we prove modularity of such representations, thus solvingSerre’s conjecture (cf. [15]) for them; non-existence follows from the factthat S 2 (1) = { 0 } .We will prove the result for p> 3, the case of p =3wassolvedbySerre(cf. [16, pag. 710]), based on a previous analogous non-existence result ofTate for the case p = 2 (cf. [18]). In fact, in our proof we will at some stepneed the validity of the result for