LetR be an integral domain andI a non-zero ideal ofR. The canonical mapR→R/I is called atorsion-free cover ofR/I if everyR-homomorphism from a torsion-freeR-module intoR/I can be factored throughR. The main result of this paper is thatR→R/I is a torsion-free cover if and only ifR is complete in theR-topology andI is an ideal of injective dimension 1. In this caseI is contained in the Jacobson radical ofR. And if Λ is the endomorphism ring ofI, then Λ is a quasi-local domain. IfI is a flatR-module, thenQ→Q/Λ is a torsion-free cover, whereQ is the quotient field ofR. And thenQ/Λ is an indecomposable injectiveR (and Λ) module. Special results are obtained ifR is a Noetherian domain or a Prufer domain.