Nearly all balloons that have been constructed have been for flights from the surface to altitude. That requires a structure that can survive tethered at the surface in a range of wind speeds in high-density air, then expand to several times their original size in order to maintain lift as the air density decreases. The minimum density ends up being determined by the need to survive liftoff.

ATO is a two-stage system with the upper stage being constructed at 140k feet. Given that it will never be tethered to the ground, it will experience no wind at all, because the Dark Sky Station will be in exactly the same air the upper stage is in. That means the lower limit on its density ends up being determined by other factors such as differential air speeds at different parts of the structure. That could still be a killer, but "the thinnest balloons" is not relevant here, because the thinnest balloons anyone would have experienced must still be far denser than what you could build at 140k feet.

Lift is a function of drag. At any altitude where drag would prevent the upper stage from achieving orbital velocity for that altitude, the upper stage would continue to generate lift. This profile continues smoothly until the upper stage is close enough to orbital velocity that it no longer matters. It's possible that, at whatever altitudes you'd want to operate, the upper stage would need to generate thrust continuously, and anything you wanted in actual orbit would need an extra kick, but with solar-powered ion propulsion, that doesn't seem like much of an inconvenience.

ATO will run into plenty of challenges, but I'm not aware of any that are obvious killers. There are problems like radiation and the logistics of construction in full gravity and extremely low pressure, but I imagine the DSS and upper stages will end up being constructed by robots. Even if it only ends up being used for cargo, a "grown up" spacefaring society should be sending humans and cargo up separately anyway, because they have different requirements for speed and safety. Humans can wait on the ground until the cargo they work with is safely in orbit.

Nearly all balloons that have been constructed have been for flights from the surface to altitude. That requires a structure that can survive tethered at the surface in a range of wind speeds in high-density air, then expand to several times their original size in order to maintain lift as the air density decreases. The minimum density ends up being determined by the need to survive liftoff.

ATO is a two-stage system with the upper stage being constructed at 140k feet. Given that it will never be tethered to the ground, it will experience no wind at all, because the Dark Sky Station will be in exactly the same air the upper stage is in. That means the lower limit on its density ends up being determined by other factors such as differential air speeds at different parts of the structure. That could still be a killer, but "the thinnest balloons" (mentioned in Charles Pooley's answer, as being barely lighter than the 4 g/m³ of air at that altitude) is not relevant here, because the thinnest balloons anyone would have experienced must still be far denser than what you could build at 140k feet.

Lift is a function of drag. At any altitude where drag would prevent the upper stage from achieving orbital velocity for that altitude, the upper stage would continue to generate lift. This profile continues smoothly until the upper stage is close enough to orbital velocity that it no longer matters. It's possible that, at whatever altitudes you'd want to operate, the upper stage would need to generate thrust continuously, and anything you wanted in actual orbit would need an extra kick, but with solar-powered ion propulsion, that doesn't seem like much of an inconvenience.

ATO will run into plenty of challenges, but I'm not aware of any that are obvious killers. There are problems like radiation and the logistics of construction in full gravity and extremely low pressure, but I imagine the DSS and upper stages will end up being constructed by robots. Even if it only ends up being used for cargo, a "grown up" spacefaring society should be sending humans and cargo up separately anyway, because they have different requirements for speed and safety. Humans can wait on the ground until the cargo they work with is safely in orbit.