Per Jim (listed as a Dev) on the Sourceforge Open Broadcaster Software Forum, OBS does not support fullscreen mode, but it is a planned future improvement.
Currently, fullscreen games can't be captured via software means without using DXTory or something.
Fortunately, I do have plans in the future to add fullscreen game capture, but because it's a rather difficult problem to tackle and will require a lot of time and focus, it's a bit lower on the priority list and won't be in for quite some time.
DXtory looks to be a non-free screen scraping application, and I'll go out on a limb and assume that's not what your looking for, as it would be a pretty advanced technical solution.
Others in the thread suggest borderless fullscreen windowed mode as a workaround. This would accomplish many of the goals you are shooting for, and would get around the OBS limitation. If you follow that link there are a number of applications listed, including Shift which was mentioned in comments. There is also some mention of potential drawbacks of each, such as compatibility with Punkbuster.
Finally, as this PC Gamer article points out, be sure that your source content is not minimized when you experience "black screen".
You can now preview the stream or start broadcasting, but you won’t see anything if the game is minimized
If you are willing to entertain other options, this video describes how to setup Xfire and Twitchtv for fullscreen mode games, and towards the end he shows proof that it works with Fifa '12. Both are free services, but with Twitchtv your viewers may have to sit through some ads before watching, if they don't want to pay for the ad free service.