Hey, Lyoness - first off all, congratulations!!!! That's FANTASTIC news, YAY you!
And I guess in response to your question, here's my thinking on it: you already know you are not accepting their offer. What will it change in terms of your telling them that, if you wait several weeks up to a month to do it, or do it right away? In the end, you're still not planning to attend. Trying to see it from the perspective of a graduate studies director, I guess I'd have to say I'd prefer to know sooner rather than later, especially when budgets are tight, because maybe I have several really promising candidates, but I can't offer anyone anything until I know what the cohort looks like.
I don't think it looks ungrateful. I think it looks like you got a better offer and you are considerate enough to let me know up front rather than sitting on our offer and ultimately rejecting it.
Actually, from the perspective of someone in the department, I think I might resent the candidate who doesn't reply right away and then rejects the department's offer a month or two later. If I offer you an admit with funding in January, and you wait until April 15 before notifying me you aren't going to accept it, you've screwed my department out of some potentially very good applicants who maybe are on my waitlist but have since accepted elsewhere. Because we haven't heard from you in the form of turning down the offer, we probably think you are coming at that point, so it's a rude shock to find that position still open at the end of the admissions season. There may even be department wide or school wide ramifications for it in terms of funding - maybe, since you reject the position at the end of the admissions season and we have already notified all of our candidates, our department will end up short a doctoral student next year and that funding will be applied elsewhere. I could see that scenario play out in combo departments, like English/Comp Lit, or Romance languages, or something like that, where funding has already been cut or programs are threatened - if a Dean wanted to, s/he could make a case for "Well, this-or-that department didn't even attract a full cohort this go-around, so they maybe don't need five student slots a season, next year they'll just get four and we can put that moeny towards funding a position in another department where there are more candidates applying", etc. etc. I don't know. It could happen. I know that at a university I attended for Master's level work in French, there is no longer a Master's program in French, because they didn't have enough admits to the program to justify funding those students anymore, so it just...went away.
Maybe we have the thinking messed up. Maybe it's best to try to respond within ten days or so, if we can...? I do think in the end, it would be the better way to proceed for everyone else involved, and I don't think a department will hold it against you if you made the decision to apply elsewhere, got accepted, and decided to accept that offer - I mean, it happens every year, right? They know you have likely applied to several programs.