The elements of O(d, d, Z) are shown to be discrete symmetries of the space of curved string backgrounds that are independent of d coordinates. The explicit action of the symmetries on the backgrounds is described. Particular attention is paid to the dilaton transformation. Such symmetries identify different cosmological solutions and other (possibly) singular backgrounds; for example, it is shown that a compact black string is dual to a charged black hole. The extension to the heterotic string is discussed.
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We would like to thank R. Dijkgraaf, C. Nappi, E. Verlinde, E. Witten and B. Zwiebach for discussions. The work of AG. is supported in part by DOE grant DE-FGO2-90ER40542, and that of MR. is supported by the John Simon Guggenheim Foundation.