A concept of CP-violating (T,P-odd) permanent molecular magnetic moments µ CP is introduced. We relate the moments to the electric dipole moment of electron (eEDM) and estimate µ CP for several diamagnetic polar molecules. The moments exhibit a steep, Z 5 , scaling with the nuclear charge Z of the heavier molecular constituent. A measurement of the CP-violating magnetization of a polarized sample of heavy molecules may improve the present limit on eEDM by several orders of magnitude.PACS numbers: 11.30. Er,32.10.Dk,31.30.Jv It is a common knowledge, that heteronuclear diatomic molecules possess a static electric dipole moment aligned with the internuclear axisn, D = Dn. For a diamagnetic molecule, however, there is no similar magnetic moment. As demonstrated below, an existence of such a magnetic moment would violate both paritytransformation (P) and time-reversal (T) discrete symmetries. Because of the compelling CPT theorem, an observation of this magnetic moment would provide an evidence for the CP violation [1,2]. CP-violation, although observed in particle physics, still remains a mystery, as much stronger CP-violating mechanisms may be required to explain the matter-antimatter asymmetry of the Universe.Here we introduce the molecular CP-violating magnetic moments, µ CP = µ CPn . We propose a measurement of µ CP via detection of ultra-weak magnetic fields generated by a polarized sample of diamagnetic molecules. For several molecules we evaluate µ CP and express them in terms of the permanent electric dipole moment (EDM) of electron, d e . A measurement of nonvanishing molecular CP-violating moments would reveal the elusive electron EDM (eEDM) (EDMs violate both T and P symmetries). While no EDMs have been found so far, it is anticipated that the next generation of experiments may finally discover the EDMs. Indeed, most supersymmetric extensions of the Standard Model of elementary particles predict eEDMs that are within a reach of planned and on-going experimental searches (see, e.g., a popular review [3]). The present limit on d e comes from an atomic Tl beam experiment [4], d e (Tl) < 1.6 × 10 −27 e · cm .(1)Here we propose an experimental search for the CPviolating magnetic moments of heavy polar molecules. We argue that the limit on µ CP derived from such experiments would imply constraints on d e that are several orders of magnitude better than the present limit (1). In principle, the experiments can be carried out with any diamagnetic polar molecules. However, there is no particular advantage in using polyatomics, and we restrict our consideration to polar diatomic molecules.In the remainder of this paper, unless specified otherwise, we use atomic units |e| =h = me ≡ 1 and the Gaussian system for electro-magnetic equations. In these units, the Bohr magneton is µB = α/2, where α ≈ 1/137 is the fine structure constant, and the unit of magnetic field is me 2 e 5 /h 4 ≈ 1.72 × 10 7 Gauss.General considerations. Diatomic molecule is characterized by the projection Ω = (J ·n) of the total electronic angu...