Registry keys need to be edited to change the default browser.
I have found these keys so far and they seem to do what I want, but I am not sure if I have found all of them:
Data in:
HKEY_CURRENT_USER\Software\Classes\http\shell\open\commandHKEY_CURRENT_USER\Software\Classes\https\shell\open\commandHKEY_CURRENT_USER\Software\Classes\ftp\shell\open\command
Value in:
HKEY_CURRENT_USER\Software\Classes\Local Settings\Software\Microsoft\Windows\Shell\MuiCache