One month at work with no work to do

What kind of work do you do now? If you're looking for investment management experience you could ask for a project related to finance. If your company doesn't have anything like that for you do to you could start studying up on modeling.

But from my experience, I don't think one month of doing some kind of investment management-related project will make too much difference in your ability to get an internship or not. It's more about what you do when you get to b-school and how you can spin your previous experience to match what you want to do. I saw several classmates with no finance experience get finance internships and then convert those into jobs.

Good luck.

I agree with what AmyCope mentioned above. 1 month of internship won't make or break things from career perspective. If you want to utilize your time before the school starts then in my opinion start working on things such as:
1. Resume: Career mgmt. at your school is going to help you with this but it would be like a mad rush from the day one about resume, cover letter besides all the course load and signing up for classes, adjusting to a new environment etc. So, better get a head start on this important thing.

2. You can also start making a list of the companies where you want to work. I made an excel for all the companies hiring and the ones I was interested in.

3. You can start "networking" - the most imp thing to find a job from a b school. Finding alumni from your school / previous connections on LinkedIn and connecting with them. I made an excel for this as well and kept their names and email addresses (found after connecting through linkedIn), dates I emailed them, days since last contact, follow-ups, any comments etc. This is what took most of my time and is by far the most imp thing to find an internship or job.

Just my two cents...