To get something into a stable orbit you have to get it high enough to avoid significant atmospheric drag. In practice this requires an altitude of at least 200km [1] - although even at this altitude an orbit will decay fairly rapidly. The IIS orbits at 400km and most remote-sensing satellites are at 500km+.
Contrast this with Helium balloons, which usually top-out at an altitude of 35-50km with the record being 53km [2].
To stay in low-earth orbit you also need an orbital velocity of 7.8km/s. A balloon launched at the equator has about 0.5km/s horizontal component of velocity due to the earth's rotation and will not gain any as it lifts.
What you could do is use a balloon to carry a rocket up to its height ceiling before igniting [3]. This saves on fuel, but there are safety issues caused by the non-steerability of the balloon.
One advantage of launching a rocket from high altitude is that the optimal rocket nozzle shape and size changes substantially depending on the ambient air pressure. For example, the Space Shuttle Main Engine is about 25% more efficient in vacuum than at sea level.
In hindsight, I really could have worded my original question better. It didn't even occur to me that the way I worded the question implied maintaining its position, my bad.
I was just curious if you could get the balloon to maintain an altitude within low earth orbit, without concern for where the balloon actually drifts.
I was imagining a way to get the balloon to behave as a buoy at sea (not anchored), floating above the thicker atmosphere the same way a rubber balloon or raft would drift at sea.
There was a plan to get something into orbit by lifting in a balloon, then putting in a vehicle that was a cross between a wing and a balloon where as it accelerates, it generates lift, pushing it up so that drag becomes less, etc, eventually launching cheaply. But I haven't heard of the company in a while.
That doesn't sound plausible but I could be wrong.