Here's another way to do it using mysqldump utility:

mysqldump --add-drop-table -c -uUSERNAME -pPASSWORD YOURDB Links > yourdb.txt

This will dump the contents of the Links table into a file called yourdb.txt. You can then edit the file and change the order of fields by editing the create table statement (should just be a matter of cut and paste some rows into different order).

Then just do a:

mysql -uUSERNAME -pPASSWORD YOURDB < yourdb.txt

and it will do the changes. You should then do a re-sync in the admin and you are set.

Cheers,

Alex