You don't have to do 1 project in one sprint. If you have a product backlog your product owner and scrum master can place as many stories in a sprint as they want. If a sprint takes up 2 weeks of work, you should plan 2 weeks of stories in that sprint.
Next sprint, you do the same. If you don't have multiple short projects you'll probably end up with half a sprint, but in that case there's probably something else which is wrong in the company (hint: sales should sell more).
I've been in a couple of sprints where I was working on 8 projects. It's not an ideal situation, but if a 'project' only takes up 8 to 16 hours you'll have to do a lot of them. Of course the projects were only 1 or 4 stories. All of them were noted on the product backlog and went through the complete iteration.
That said, Scrum (like any other methodology) only offers some guidelines. You can take whatever you want from it, as long as you know what you pick and why you pick it. If you think you need iterations of a week, please do (I do think that's a bit fast, but some books do write about 1 week iterations).
Do note, I'm not a scrum master and have worked very little with it, but have seen some theory and did a couple of iterations with it. Someone more experienced with Scrum could possibly tell a lot more about the subject.