Been there too BrownCow1! My ds2 was a little older, about 14/15 I think. We were really worried he had turned into a lying, thieving petty criminal. He has a Saturday job now, and although the stealing has pretty much stopped, I think it's mostly because we NEVER leave money around. I never have cash in my purse or around the house.
It's a really, really horrible feeling that you can't trust your own son. I have an older ds, and he would never of dreamt of deceiving us that way. I believe it's relatively common, but it doesn't feel good. My ds2 has been brought up in a very loving, caring family but I think, he thinks he can help himself to anything so long as it's around the house - he helps himself to aftershave, chocolate, clothes too!
Anyway, once we were able to catch him, which took a while (he denied stealing till he was blue in the face), we had many long conversations etc about doing the right thing. He confided that he simply didn't have enough pocket money to compete with his mates (who all seem to have an unlimited supply of cash!), and that he knew we wouldn't give him anymore. I think he was getting about £10-15 a week at the time.
Now he has a job (which won't help you, as your ds is only 13) he has to manage his own money better. He spends it like crazy as soon as he has any, then runs out, which is why I still continue to not have cash around.
I would suggest he maybe gets a little more pocket money but he would have to earn it by doing little jobs for you. Do you know what he's mostly spending it on? I'm pretty sure my ds2 secretly smokes and ciggies aren't cheap (that's a whole other story!). I think it's something that takes a while to grow out of, and he sounds quite hormonal and teenagery at the moment! I wouldn't leave money around anymore - it's a pain, but I've got very used to hiding cash in the house and only having credit cards in my purse. Good luck and don't worry too much - I think it's a horrible phase they go through. x