The embeddability of a straight cosmic string in a Friedmann-Robertson-Walker (FRW) universe is examined. Although previous suggestions that an exact embedding for a string with longitudinal tension equal to energy density is impossible are substantiated, it is shown that the deviations of either the external metric from the exact FRW metric or of the internal structure of the string from the exact tension equals energy density are expected to be very small, of the order of the square of the ratio of the string diameter (or the evacuated shell around the string) to the Hubble radius. Thus the lack of an exact mathematical embedding leads to negligible physical consequences. The problem with solving for an exact embedding of a string in the manner of the Swiss-cheese model is examined in detail, and it is shown that the metric in the evacuated region around the string is unique. That metric is determined to lowest order in the ratio of the evacuated region over the Hubble radius. The implications of this uniqueness for the Swiss-cheese embedding of a string are discussed.