If you only have access to one DS, the only type of trading you can do (ad long as it's an Original Fat DS, or DS lite) is migration.
You put any of the following games in the GBA slot:
Than you go to your Diamond, Pearl, Platinum, Soul Silver, or Heart Gold, and on the main menu on the bottom will say "Migrate from (whatever game you put in dual slot)"
If you wanted to trade as in trade regularly, like to evolve a Pokemon, than no. It is impossible.
I've heard people using some glitch on GTS to evolve Pokemon (as long as it was Generation IV).
But I don't know how to do it so I can't describe it to you dice I have access to two DS's.
I hope this helps! ~