If it is an AT&T or compatible phone, most likely yes. If you have any problems you can get a replacement SIM card at an AT&T store.
If the HTC phone no longer charges because the battery is old, it can be replaced inexpensively. I have purchased batteries online for several cell phones. ( I don't know if the HTC has a user changeable battery ). Or it can be sent in for repair.
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Thanks! I plan getting the HTC one fixed or replaced, but I have to send it off and will be without a phone for a week or so. So I was hoping I could just get a cheap phone that will use my SIM card while I waited.