As mentioned above, it really depends on how much in the lurch you’d be leaving your employer. Where I work, anytime anyone leaves it’s a serious burden, and the more notice the better, but the difference between one week and two is rarely that significant. (Two weeks isn’t enough to complete your current project, but one week should be enough to transition to a new worker, if one is available – and having two weeks might not make a huge difference in the latter regard.)
Frequently, here, people give management plenty of notice but management tells them not to mention it until just before leaving. I don’t like that, though I understand the reasons for it.)
I once gave my work a year’s notice that I’d be gone for six months, traveling. That worked out well, and I worked for them again after I returned (but not right away, not until they had new contracts coming in, which was fine.) They definitely appreciated the advance notice.
If you do decide to give only one week notice, I suggest tellng the truth to your boss: you want/need a week off between. Since you probably won’t be able to take vacation right away in the new job, you’ll really want that time off now.
A similar thing happened to me: I was offered a job; they asked me when I could start and I said I only needed to give two weeks notice and could start immediately after that. But before it was final, my wife pointed out that if I postponed it two more weeks, we could have a vacation and it would span Labor Day, after which I could start. I simply asked the new employer if I could postpone two weeks and they had no problems with that.
Maybe you could try this with your new employer: “I won’t want to take vacation right away after starting, but I need to have a vacation this year. However, I won’t be able to do that and give my current employer two weeks’ notice. Can we delay my start by one week?” As a hiring manager, I’d rather agree to that than to get a burnt-out worker, and I’d learn that the new hire was someone who takes responsibilities seriously.