A scheme is proposed to exchange the transverse and longitudinal emittances of an electron bunch. A general analysis is presented and a specific beamline is used as an example where the emittance exchange is achieved by placing a transverse deflecting mode radio-frequency cavity in a magnetic chicane. In addition to reducing the transverse emittance, the bunch length is also simultaneously compressed. The scheme has the potential to introduce an added flexibility to the control of electron

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... and to provide some contingency for the achievement of emittance and peak-current goals in free-electron lasers.