This came up on the old boards -- I believe the overwhelming consensus was that the Con drain "officially" occurred *right before* the Weasel's initiative came up again the next round.
So the timing of the Con drain of Weasels and Stirges is kind of like the timing of casting a full-round spell like Summon Monster I. The effect goes off *just* before you move next.
And if you ask me about hasted Weasels, I'm afraid I'll have to kill you.