A bright prominence associated with a coronal mass ejection (CME) was seen erupting from the Sun on 9 April 2008. This prominence was tracked by both the Solar Terrestrial Relations Observatory (STEREO) EUVI and COR1 telescopes, and was seen to rotate about the line of sight as it erupted; therefore, the event has been nicknamed the Cartwheel CME. The threads of the prominence in the core of the CME quite clearly indicate the structure of a weakly to moderately twisted flux rope throughout the field of view, up to heliocentric heights of 4 solar radii. Although the STEREO separation was 48 degrees, it was possible to match some sharp features in the later part of the eruption as seen in the 304 {AA} line in EUVI and in the Halpha-sensitive bandpass of COR1 by both STEREO Ahead and Behind. These features could then be traced out in three-dimensional space, and reprojected into a view in which the eruption is directed towards the observer. The reconstructed view shows that the alignment of the prominence to the vertical axis rotates as it rises up to a leading-edge height of approx 2.5 solar radii, and then remains approximately constant. The alignment at 2.5 solar radii differs by about 115 degrees from the original filament orientation inferred from H{alpha} and EUV data, and the height profile of the rotation, obtained here for the first time, shows that two thirds of the total rotation is reached within approx 0.5 solar radii above the photosphere. These features are well reproduced by numerical simulations of an unstable moderately twisted flux rope embedded in external flux with a relatively strong shear field component.