I found a way to get rid of this problem but I'm not sure it makes sense (at least to me). Here it is:
I put all of this code into a seperate .JS file and specify the full
path of Autocomplete.aspx file. For example;
...
$("#txtSearchBox").autocomplete("http://localhost/MySite/
Autocomplete.aspx",
...
There must be something else about these FF vs IE issues.
Any further contributions would be appreciated.

