Detailed spectroscopic studies of metal-poor halo stars have highlighted the important role of carbon-enhanced metal-poor (CEMP) stars in understanding the early production and ejection of carbon in the Galaxy and in identifying the progenitors of the CEMP stars among the first stars formed after the Big Bang. Recent work has also classified the CEMP stars by absolute carbon abundance, A(C), into high- and low-C bands, mostly populated by binary and single stars, respectively. Our aim is to determine the frequency and orbital parameters of binary systems among the CEMP-s stars, which exhibit strong enhancements of neutron-capture elements associated with the s-process. This allows us to test whether local mass transfer from a binary companion is necessary and sufficient to explain their dramatic carbon excesses. Eighteen of the 22 stars exhibit clear orbital motion, yielding a binary frequency of 82+-10%, while four stars appear to be single (18+-10%). We thus confirm that the binary frequency of CEMP-s stars is much higher than for normal metal-poor giants, but not 100% as previously claimed. Secure orbits are determined for 11 of the binaries and provisional orbits for six long-period systems (P > 3,000 days), and orbital circularisation time scales are discussed. The conventional scenario of local mass transfer from a former AGB binary companion does appear to account for the chemical composition of most CEMP-s stars. However, the excess of C and s-process elements in some single CEMP-s stars was apparently transferred to their natal clouds by an external (distant) source. This finding has important implications for our understanding of carbon enrichment in the early Galactic halo and some high-redshift DLA systems, and of the mass loss from extremely metal-poor AGB stars. Abridged.
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