What are your thoughts?

So I did not think I was going to be in the position and I am very grateful that I am. Even though my original hopes of an S&T summer internship looks like it is not going to work out, I was hoping to get a few opinions of some of you. My situation is this; I was just offered a summer internship position at The Hartford in investments/risk management. I was able to network my way in and by pass the second round phone interview and in person interview. I know the women now on a personal level after shadowing her for an entire day and I definitely like her which makes things a bit more difficult. So here is where my issue begins. I have to tell her by the end of next week on if I want the job or not. I have a second round interview next Friday with a company I am modestly interested in. Now on top of that, I had a phone interview with a company today that I really like and the position is exactly what I'm looking for (portfolio management intern) but I will not know if I get a second round by possibly next Friday or possibly the Friday after that. So with that what would your suggestions be? I am thinking about asking the company I interviewed with today to try to speed up the process since they are my number one company or just accept the job at the Hartford and if I get offered a job at the other company and just tell them I do not want it anymore?

I am extremely excited I was able to get such a string of luck after so many denials in a row, but was hoping to get a second opinion. Thanks in advance for your help!