Depending on how short is short, they may let you take annual leave to do it, or maybe unpaid leave. This is a good opportunity to see if your new employer is willing to be flexible and reasonable. I would hope they would be glad to see that you have the integrity to want to meet a prior commitment, even when it involves some inconvenience to you.
I think it would have been best to mention it at interview, but if you give them notice of it immediately then it will give them time to plan around it.
I aqree with padraiq that you should have said something when you accepted the position. What you have to do is let your superior know now. Don't put it off any longer.