You could create an identical application and redirect users to this by using a DNS change or a simple redirection in IIS before you make a change.
You mention internal users suggesting that the app is only used internally? If so you could force a DNS change on your domain/network to the new app and they will never know the difference. In the meantime make the change to your 'live' app and then do the reverse DNS change back again.
Failing that you could always perform the change out of hours. Every application requires a maintenance window every now and then.