Yes Kaname did give Artemis to Yuki, but Yuki woudn't be able to kill Kaname. Kaname said if he were to let her go, he would rather KILL HER HIMSELF...Or have her kill him. They both said to each other that they don't mind what they do to each other...
That part is so romantic, because, he wants to kiss yuki, but yuki dont allow him, so u can see the black shadow over him!!! so funny
and then he is going after yuki, and say that things to her, and finish kissing her!!! just like he wants!!! and later he doesn't let her go!!!