Yes the GoPro has stabilization, but it can result in undesirable footage when shaking heavily under bright light (search for "jello effect").
One way to correct the jello effect is to add ND filters before the lens to reduce the amount of light and therefore slow the shutter (since you can't otherwise control the aperture). There are filter holders that can be screwed on top of the box, or attached directly to the lens when the GoPro isn't in its box.
The other way is to re-stablilize in post-processing; results can be good to very good depending on the source footage. Heavy shaking and big jello can't really be corrected, but slow shake can be made to disappear completely.
One way to correct the jello effect is to add ND filters before the lens to reduce the amount of light and therefore slow the shutter (since you can't otherwise control the aperture). There are filter holders that can be screwed on top of the box, or attached directly to the lens when the GoPro isn't in its box.
The other way is to re-stablilize in post-processing; results can be good to very good depending on the source footage. Heavy shaking and big jello can't really be corrected, but slow shake can be made to disappear completely.
This was shot with a Tarot gimbal attached to a bike, and no software stabilization: http://footage.shutterstock.com/clip-5460407.html
This was also shot with a gimbal but (re-)stabilized in post: http://footage.shutterstock.com/clip-5317121.html
both with a GoPro 3 Black Ed.