We present a method for measuring electrical currents enclosed by flux rope structures that are ejected within solar coronal mass ejections (CMEs). Such currents are responsible for providing the Lorentz self-force that propels CMEs. Our estimates for the driving current are based on measurements of the propelling force obtained using data from the LASCO coronagraphs aboard the SOHO satellite. We find that upper limits on the currents enclosed by CMEs are typically around 1010 Amperes. We estimate that the magnetic flux enclosed by the CMEs in the LASCO field of view is a few ×1021 Mx.