Your Balance account is accessible across multiple devices. As long as you are signing in as an existing member on your new device with the same email and password as that on your old device, you'll be able to restore your account and progress.
To log in as an existing member you'll want to download the app to your new device, open the app, and tap on the text that says, "Already a member? Sign In." Signing in with your existing account will restore your purchase and progress.