In the ¯rst Heisenberg group H1 with its sub-Riemannian structure generated by the horizontal subbundle, we single out a class
of C2 non-characteristic entire intrinsic graphs which we call strict graphical strips. We prove that such strict graphical strips have
vanishing horizontal mean curvature (i.e., they are H-minimal) and are unstable (i.e., there exist compactly supported deformations for which the second variation of the horizontal perimeter is strictly negative). We then show that, modulo left-translations and rotations about the center of the group, every C2 entire Hminimal graph with empty characteristic locus and which is not a vertical plane contains a strict graphical strip. Combining these results we prove the conjecture that in H1 the only stable C2 Hminimal entire graphs, with empty characteristic locus, are thevertical planes.