Presumably the border you're referring to is the one that only appears while you have the bibliography selected. That belongs to the bibliography building block and neither displays unless the bibliography is selected nor prints.

The only way of getting rid of that is to delete Word's bibliography building block - and your bibliography with it. You could, of course, insert a bibliography field and your own bibliography heading, eschewing the use of Word's bibliography building block.

__________________
Cheers,
Paul Edstein
[Fmr MS MVP - Word]