A new relevant class of perturbative models involving elementary scalars is proposed in which the theory simultaneously satisfies the Veltman conditions and is conformal at the classical level. These are so-called perturbative natural conformal (PNC) theories. I will show that PNC models are very constrained and thus highly predictive. Among several PNC examples, we have discovered a remarkably simple PNC extension of the standard model in which the Higgs is predicted to have the experimental value of the mass equal to 126 GeV. This model also predicts the existence of one more standard model singlet scalar boson with a mass of 541 GeV and the Higgs self-coupling to emerge radiatively. Other PNC examples generally predict a somewhat smaller mass of the Higgs. These results can be a useful guide when building extensions of the standard model featuring fundamental scalars.