We present a method for calibrating images acquired by the Dawn Framing Camera (FC) that is based on the results of an in-flight calibration campaign performed during the cruise from Earth to Vesta. We describe this campaign and the data analysis in full. Both the primary camera FC2 and the backup camera FC1 are radiometrically and geometrically calibrated through observations of standard stars, star fields, and Solar System objects. The calibration in each spectral filter is accurate to within a few percent for point sources. Geometric distortion, small by design, is characterized with high accuracy. Dark current, monitored on a regular basis, is very low at flight operational temperatures. Out-of-field stray light was characterized using the Sun as a stray light source. In-field stray light is confirmed in narrow-band filter images of Vesta. Its magnitude and distribution are scene-dependent, and expected to contribute significantly to images of extended objects. Description of a method for in-field stray light correction is deferred to a follow-up paper, as is a discussion of the closely related topic of flat-fielding.