I don't know about the Go-Pro, but I have a Contour Roam which came with a rather sparse set of mount options. However, I have fixed up the following mounting on the stem, which could equally be inverted so that the camera is below the bars. A similar set-up could be used for bar mounting.
The tapering pillar fixed to the stem is the bottom part only of a Topeak Bar Extender
which I've had for years but don't use any more. I had to drill out the holes slightly to take a 1/4" UNC bolt which screws straight into the tripod bush of the Contour. The nuts and washers on the bolt are for spacing.
Since the Contour can be rotated to any angle within its casing, it's always possible to get a horizontal image whatever the position.
The main difficulty was finding a 1/4" UNC (or BSW) bolt. Why the camera industry has stuck to non-metric thread sizes I'll never know (probably US influence). Anyway, don't try using a metric bolt - none of them will fit. And a lot of the DIY places now sell only metric nuts and bolts. I had to rummage through my father's old collection, which luckily I've kept...
One slight disadvantage is that I need to carry a screwdriver if I want to de-mount the camera. However, it's reasonably thief-proof in this set-up.