Paul Mccobb chairs repair  

I don't have any eperience with that particular chair but I would just glue and clamp it. 
Do you have any bar clamps?
Make sure the joint is clean before clamping. I'd gently clean the crack if there are any debris inside. be sure not to disturb the wood grain or when the crack is clamped it may not line up properly. 
Maybe do a dry clamping before glueing to see how she looks. If all looks good apply some wood glue with a brush and maybe some type of thin ridged plastic to get into the narrower part of the crack. Two or three clamps should be good. 
Once clamped the squeezed out glue will be easily removed with some Damp paper towels. Make sure to remove any excess glue before it dries. 

It would be interesting to try test clamping it before you put the glue in.  The interesting thing would be whether the two sides bend back together, or whether they have so strongly bowed away from each other that you can't bully them back together.  
And use some soft wood blocks or tick rags or something between the clamp jaws and the chair so you don't leave clamp dings.  
If it squeezes back together, you could proceed to glue it and clamp it per great white's very good instructions, BUT, it would be a much stronger joint to completely break it apart, then re-glue.  If you don't completely break it apart there will be debris and bent splinters in the joint, which will make a wider, weaker glue line, and you won't be able to get glue in quite a lot of the root of the crack.
If it doesn't squeeze back together, you have a whole different sort of problem, which will be much harder to fix.
Note it is slightly possible that test clamping the chair might break the crack open all the way, and this would be a good thing for repairing it.