Ligate the two at equimolar ratio (relatively high concentration of DNA) for a short time to prevent excessive polymerization. Kill the ligate by heating.
Either gel purify the resulting product or at least precipitate the DNA to clean it up.
Fill the ends by DNA polymerase or a single cycle of PCR using a good quality error-correcting enzyme, such as PFU polymerase. Clean up the DNA again.
Perform the ligation under conditions that favors blunt end-to-end ligation (low concentration.
Some of your product will have one end ligated with AT overlap and the other with blunt end.
If preserving one AT overlap is not cricial, fill all ends and do a single blunt end ligation.
1. Remove the overhang by mung bean nuclease (or T4 DNA polymerase, or any other DNA blunting enzymes).
2. Purify your blunt end DNA fragment.
3. Incubate your purified DNA fragment with dATP and Taq DNA polymerase at 72"c for 1 minute (Taq DNA polymerase catalyzes the non-template directed addition of an adenine residue to the 3´-end of both strands of DNA molecules).
4. Purify your DNA fragment and ligate it onto your pGEMT vector.