Today we're going to do a very quick, but functional, repair on a broken hand caned seat in a canoe. Although this is not the correct way to repair these seats, it definitely works and will get you through until you can buy a new seat or properly hand cane the broken one.
I'll mention, I'm a pretty big guy (6', 220lb) and this seat has no issues holding me up. I needed this canoe for a camping trip in two days, so I had no time to order cane or a seat, but with some rope and an hour or two of time, we were good to go.

Step 1: Introducing the "snowshoe" Weave.

Starting with a broken cane seat, we're going to use rope to fashion a new seat using the snowshoe weave. This is the same weave used on some vintage canoes such as E.M. White and, you guessed it, snowshoes (both using rawhide of course). You can obviously use this to build snowshoes, and we may do that in the future.

Step 2: Cleaning Up the Broken Seat.

Starting with your broken seat, you'll need to cut away as much of the cane as possible. Depending on whether the seat was hand caned (holes drilled around perimeter), or a premade panel (routed channel with a strip glued in), you may need to use a flat head screwdriver or small chisel to pop out the strip holding it in. Try not to destroy the channel if you plan to properly fix the seat in the future. My canoe will be getting brand new seats front and rear, so I didn't really care at this point.
The photos here do not show the seat cleaned completely, but subsequent steps will.

Step 3: Materials and the Weave.

Go to your local hardware or department store and buy a 100ft. coil of inexpensive cotton clothesline. Traditionally this would be done with rawhide, but since I had less than 48 hours before we were leaving, I opted for a more readily available medium. On an average seat you'll use probably 70-80 feet. For a more permanent fix, you could use rawhide (will require extra steps such as soaking, and the extra time for finished product to dry), poly rope, or even paracord.
Copy, save, resize, print, whatever you have to do, the diagram photo included with this step. It's invaluable, and much easier to look at on paper.
You may also want to procure a pair of gloves, pulling on the rope as tight as you need to can be a chore.

Step 4: Starting Your Weave.

Tie a simple overhand knot in one end of your line, this will act as a stop. Using a clove hitch, attach the line to the place marked "start" in the earlier diagram. Following the diagram exactly, begin your weave, being sure to pull the rope as tight as you can with each step.

Step 5: Finished Seat.

Following the diagram, continue weaving, making sure everything is pulled tight until you have finished your new seat. Tie in another clove hitch as you did at the start, and don't forget the overhand knot as a stop. As was mentioned, this isn't a permanent fix but in a bind it works great. Happy paddling!