There are a few ways you could tackle the problem.
The simplest is to dig a diversion so that the river can flow to the map edge while staying within the same biome (so it all freezes at the same time). You could also dig under the freezing biome, since the temperature underground is always above freezing. You won't be able to dig the tiles right at the map edge so you'll need to designate them to be smoothed and then carve fortifications to allow the water to escape.
Another simple option is to dam the river upstream of the waterfall. If you use a raising bridge then you can link it to a lever and toggle it on and off seasonally. Since the whole river freezes this is very easy to do. Just make sure you raise the bridge before the water on top of it freezes, otherwise it will probably destroy your bridge.
If you want to keep the waterfall going during winter another option is to add an overflow drain to the canyon -- either a tunnel leading to the map edge or a minecart
portable drain. The engineering involved in this approach is a bit more complicated, so for notation we'll say the bottom of the canyon is z0 and your fort's entrance is currently at z1. Build a wall across the canyon on z1 between the waterfall and your entrance, above but not blocking the water at z0, then put the portable drain on the waterfall side of the wall at z1.
The steady state side view looks like this, where # is a natural or artificial wall, numbers are water, E is your entrance, and D is the drain:
z4 1234567777777
z3 1############
z2 1#
z1 EEE #55D555#
z0 77777777777777777777#
z-1 #####################
I'm at least 80% sure that won't be able to flood your fortress.
EDIT: One more very dwarfy option is to use magma to stop the water from freezing. If there's magma 1z below the water it will stay above freezing. The only part I haven't figured out properly is how to get magma under the water at the map edge.
If you're feeling really dwarfy, both burning artefacts and caged creatures made of fire will also work as portable drains. They're too hot for your dwarves to handle though, so you can't easily move them once you've created them.