I suspect that this method would work without error if you did not use external authentication (at least it does for me). I forget the details, and I think they vary with whether you are using mac or windows systems, but there are other methods that can be used as a desktop short cut for opening the remotely hosted database one of those methods may be able to do the job without getting that extra password dialog.
Anyone reading this thread that knows how to set up one of the alternative methods?
Thank you PhilModJunk for your valuable contribution. Although you didn't give me the solution directly you most definately gave me the inspiration on how to find the solution myself. I did some research and discovered that if I create a Windows Shortcut to "fmp7://127.0.0.1/Menu" (without the quotes and with the correct IP) that it works flawlessly everytime. Thank you for your help.
Ok, it appears that this didn't solve the problem at all. I can login to the Menu without the previous login dialog / authentication issues. However, as soon as I click on any of the menu items FileMaker freezes up for a moment while it's thinking and then displays the login dialog. The behaviour is exactly as it was before I changed the open remote database desktop shortcut - meaning that I get the login dialog twice and every time I login to the computer.