I concur with neufer's recent post on the Minuteman III, that the cloud is likely from this sudden blow-out of the third-stage motor. Also according to the Wikipedia article on the Minuteman III (http://en.wikipedia.org/wiki/Minuteman_ ... LGM-30G.29
About 180 seconds after launch, 3rd stage thrust terminates.
As a previous post notes, the missile launched at 6:34:59 AM PDT, or 3:34:59 AM Hawaiian time. The above data would put the third-stage burn-out at ~3:37:59. The cloud first becomes visible in the CFHT mp4 at ~3:39:40.
A time of formation of the cloud at ~3:38:00 at third-stage burn-out is broadly consistent with the first appearance and rate of expansion of the cloud in the video. The venting from engine shut-down would have taken place over an extremely short time frame, accounting for the thinness of the shell. This would also likely not be enough time for asymmetries in the vapor distribution to form, resulting in the spherical shape of the shell.
This also accounts for the fact that there is clearly a cloud of vapor from a missile which is propelled by solid fuel (although, again according to Wikipedia, the Minuteman III does have a small liquid-fueled post third-stage motor). Similar phenomena seen previously have been attributed to fuel venting from liquid-fueled rockets or missiles (e.g., http://blogs.discovermagazine.com/badas ... ut-norway/
), but this isn't really an option for the Minuteman III.