We recently closed in a carport to make an additional room. The slab for the carport is on-grade, so now when it rains, water pours in between the sill plate and the slab. The slab extends about 1-1/2 feet beyond the wall and is covered by eaves of equal length. How should I stop this leak?

Well the way I'd approach your problem would be to:
1. Using a masonry blade (skill saw/masonry saw/grinder) score a line approximately 2 inches outside of the wall all the way through the slab.
2. Cut a second line approx. 5 inches from and parallel to the first cut.
3. Remove the concrete and install a trench drain with grate.
4. Prior to installing the grate, remove the first 5-8 inches of siding and apply exterior caulking 5 inches up the walls from the slab and apply flashing, the flashing must be Z-bent so as to be flat against the wall's sheathing, flat against the slab, and flat against the inner wall of the trench drain. Re-insall siding.
5. All drains should converge into one drainage line leading to the lowest elevation of the slab.
I know this is labor intensive but it's the surest way to stop the water. If the walls were built at the slab's edge, the flashing could have been applied prior to the sheathing or siding being installed and it would've prevented this. I hope you or your builder remembered that the walls must be 8 inches above grade by code, and that the sill/sole plate must be preservative treated wood or equal quality.