Monolayer transition metal dichalcogenides (TMD) have numerous potential applications in ultrathin electronics and photonics. The exposure of TMD-based devices to light generates photo-carriers resulting in an enhanced conductivity, which can be effectively used, e.g., in photodetectors. If the photo-enhanced conductivity persists after removal of the irradiation, the effect is known as persistent photoconductivity (PPC). Here we show that ultraviolet light (λ = 365 nm) exposure induces an extremely long-living giant PPC (GPPC) in monolayer MoS2 (ML-MoS2) field-effect transistors (FET) with a time constant of ~30 days. Furthermore, this effect leads to a large enhancement of the conductivity up to a factor of 107. In contrast to previous studies in which the origin of the PPC was attributed to extrinsic reasons such as trapped charges in the substrate or adsorbates, we show that the GPPC arises mainly from the intrinsic properties of ML-MoS2 such as lattice defects that induce a large number of localized states in the forbidden gap. This finding is supported by a detailed experimental and theoretical study of the electric transport in TMD based FETs as well as by characterization of ML-MoS2 with scanning tunneling spectroscopy, high-resolution transmission electron microscopy, and photoluminescence measurements. The obtained results provide a basis for the defect-based engineering of the electronic and optical properties of TMDs for device applications.