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We show that a Goedel-like deformation of AdS3 in heterotic string theory canbe realized as an exact string background. Indeed this class of solutions isobtained as an exactly marginal deformation of the conformal field theorydescribing the NS5/F1 heterotic background. It can also be embedded in type IIsuperstrings as a Kaluza-Klein reduction. We compute the spectrum of this modelas well as the genus one modular invariant partition function. We discuss theissue of closed timelike curves and the propagation of long strings. Theydestabilize completely the background, although we construct another exactstring background that may describe the result of the condensation of theselong strings. Closed timelike curves are avoided in that case.Comment: 23 pages; v2: minor changes, references added, version to appear in  JHE

Quantization of heterotic strings in a Goedel/Anti de Sitter spacetime and chronology protection