Let A be a local conformal net of von Neumann algebras on S-1 and rho a Mobius covariant representation of A, possibly with infinite dimension. If rho has finite index, rho has automatically positive energy. If rho has infinite index, we show the spectrum of the energy always to contain the positive real line, but, as seen by an example, it may contain negative values. We then consider nets with Haag duality on R, or equivalently sectors with non-solitonic extension to the dual net; we give a criterion for irreducible sectors to have positive energy, namely this is the case iff there exists an unbounded Mobius covariant left inverse. As a consequence the class of sectors with positive energy is stable under composition, conjugation and direct integral decomposition.