Λ-superfluidity in a core of neutron stars is studied with a realistic approach. It is found that Λ-superfluid exists in a restricted density region ρ (ρt-ρd) with ρt 2ρ0 (ρ0 being the nuclear density) and ρd (2.6-4.6)ρ0 depending on the pairing interaction and the core model. This restriction suggests that neutron stars compatible with cooling would be not so massive. Introduction Study on the superfluidity of hyperons admixed in neutron star cores is of increasing interest not only with regard to many-body problems in hadronic matter including exotic components but also to the cooling scenario of neutron stars inferred from surface temperature observations. It has been suggested that some neutron stars are cooled much more rapidly than expected from the standard cool- ing scenario (i.e., modified URCA process; e.g., n + n → p + n + e − +¯ νe, and the inverse process), and a more efficient cooling mechanism is needed. 2) In this con- nection, so-called hyperon 3) associated with neutrino emission processes as of the β-decay type including hyperons (Λ → p + e − +¯ νe ,Σ − → Λ + e − +¯ νe, etc.), provides one of the rapid cooling mechanisms. The direct action of such rapid cooling, however, leads to a surface temperature much lower than that observed. This demands some suppression mechanism, most naturally superfluidity, to control the cooling rate. Therefore it becomes of special interest to investigate whether superfluidity is possible in the hyperon-mixed phase believed to exist in neutron star cores. In this paper, as a typical example of superfluidity, we focus attention on Λ-hyperon pairing, since at present the uncertainty in the ΛΛ interaction is relatively small due to the information from various experimental data (especially, the available data from double Λ hypernuclei) and theoretical studies. Very recently, Balberg and Barnea 4) studied this subject by using an effective ΛΛ interaction ˜ VΛΛ for ΛΛ pairing interaction. However, their approach using an effective interaction for the pairing problem is not justified, as shown later. Moreover, the ˜ VΛΛ they used 5) is based on the G-matrix calculations for two Λ particles immersed in symmetric nuclear matter, but such situation is basically different from the actual one of the hyperon- mixed phase composed of neutrons as a dominant component, protons, Λ-particles and other hyperons. In addition, the effective mass they used for Λ is not suitable.