In the paper  we proved that the only stable C2 minimal surfaces in the first Heisenberg group ℍ1 which are graphs over some plane and have empty characteristic locus must be vertical planes. This result represents a sub-Riemannian version of the celebrated theorem of Bernstein. In this paper we extend the result in  to C2 complete embedded minimal surfaces in ℍ1 with empty characteristic locus. We prove that every such a surface without boundary must be a vertical plane. This result represents a sub-Riemannian counterpart of the classical theorems of Fischer-Colbrie and Schoen, , and do Carmo and Peng, , and answers a question posed by Lei Ni.