In short, yes.
But: you must do it at the same time, ensure that you transfer the money to the new account; your new provider will have a form for you to fill in ( if you withdraw it you lose the ISA status). You can invest up to your limit for the new tax year.
Transfers don't count towards your yearly limit. So if you transfer money from account A to account B you can still put new money into either account but not into both in the same tax year.