My guess is that if you're doing weight training and still eating too many calories, your arms WILL get bigger.
Fat doesn't 'turn into' muscle - you need to burn off the fat by reducing calories in your diet and burning calories by exercise.
What you'll end up with otherwise is bigger arms - muscles with a thick layer of fat over them.
Sorry...to burn fat you need to reduce your caloric intake...that means changing your nutrition and eating healthy, monitoring your calories and what kinds of foods you eat. In other words, a "diet" - although I prefer to think of it as permanent healthy lifestyle changes.
Using Jennifer Aniston as an example - I've read that she follows The Zone - that's what works for her - and likely she works out daily as well to keep herself in shape.
Are you *only* working your arms with your weight training? that could be a mistake. Your larger muscles (chest back legs) have a lot more potential to gain muscle. So make sure you weight train those muscles as well. Even without a diet, that should help you lose fat overall, which may help the appearance of your arms. With some diet modifications you'd see even more results. Remember that everyone's body is different in terms of where you carry extra weight and where you will see results first. In my opinion it's hard to spot train for appearance and get results unless one is already a low body-fat skinny type girl.
Sadly, spot reducing doesn't work or else I'd be doing stomach crunches non stop. Best bet is try to do some all over weights, add in some cardio and change your diet to get rid of the unwanted fat.