How to stop cold drafts caused by a warped door sill?
I can't tell from the picture but it seems likely that the floor beneath the middle of the sill is sagging or just lower than the ends of the sill for whatever reason.  If that is the case, and it were me, I'd pull up the existing sill and shim up the area that is too low.  Set a straight edge across the width of the door frame to see how much material needs to be added under the sill.  I would use roofing felt for that.  You can cut it with a utility knife, and it won't rot or compress very easily. Once you have filled in the low area so that no light shows beneath the straight edge, re-install the sill.  Another advantage of using roofing felt is that the screws you use to secure the sill will drive through the felt quite easily.

Here is a link to a website that discusses how to use shingles and felt to level an entire subfloor prior to flooring installation.  http://bit.ly/1Ii80Qi  The scope of that project is much larger that what you are dealing with, but the principles involved are exactly the same.