Surely the solution to the degree and no experience problem has been presented?
When I did my degree (13 years ago) I did a sandwich course. I did a maths degree and got a job for a year at Lockheed Martin. It was fantastic, I learned Visual C++ and that year (with good references) allowed me to pick up a job as soon as I left uni.
Maybe things have changed from that now, I do not know. However, the years experience that you get from that kind of course is invaluable!
Another thing I was told by a HR interviewer after I got another job was that I was interviewed because my CV contained things that I did outside of work and was successful at e.g. senior grade in martial arts and international softball player. These things show that you can get your arse in gear.