Problems in ligation. - (Aug/01/2015 )

I am trying to ligate an asymmetric adapter with one T overhang to a target with A overhang. The adapter is 5' dephosphorylated on both sides. Doe anybody know if the ligation will work, or should I have the adapter Phosphorylatyed? The target DNA is phosphorylated, so it will be a sort of single strand ligation on a double stranded DNA 

I think it must be phosphorylated. However I have heard in some cases it works without phosphorylation. 

At least one of the two strands must have a single 5' phosphate. This ligation will fail unless the target is phosphorylated, which you say it is. You can phosphorylate your primers or the annealed complex with PNK rather than re-ordering them. If you are having problems, I would try that.