Let "God" = a maximally great being (omnipotent, omniscient, and morally perfect in every possible world).
1. God's existence is either necessary or impossible. (Premise)
Implicit in (1) is that God cannot exist contingently. If God were contingent, then He wouldn't be maximally great, since it is greater to be necessary than to be contingent. The very understanding of God, then, requires that God is either a necessarily existent entity, or else impossible.
2. God's existence is not impossible. (Premise)
(2) is the more controversial premise. More on this below.
3. Therefore, God's existence is necessary. (Conclusion)
In short, it follows from (3) that God actually exists.
It remains to be seen whether (2) should be accepted. Intuitively, most of us probably grant that God's existence is at least possible; and obviously, if it is possible, then it is not impossible. However, a skeptic could easily reverse this:
(2'): God's existence is not necessary.
In support of (2'), one might appeal to the intuition that there is a possible world in which God does not exist.
One relevant question, then, is this: is the intuition of (2') as strong as the intuition of (2)? Both premises are question-begging, barring any additional argumentation that would either prove or make it likely that one premise is correct.
Perhaps the theist cannot prove that (2) is correct and that (2') is false, yet still be within her epistemic rights in believing that. This is Alvin Plantinga's position - e.g. the modal ontological argument is rationally acceptable, even if it is not a conclusive proof.
In any event, there have been some notable attempts to prove that God's existence is possible. Clement Dore and Robert Maydole immediately come to mind. Being someone who accepts the actual existence of a maximally great being (God), I'm especially interested in the developments of a proof of the possibility premise.