JabRef is capable of exporting the contents of the BibTeX database, along with groups information, to an external MySQL database.
JabRef will then connect to the specified database, drop the existing tables, create new tables, and populate those tables with entries and groups information. Note that you will not be prompted for the connection information on subsequent exports. If you would like to export to a different database, you can change the connection information by choosing File -> Connect to external SQL database (or by clicking the associated toolbar button), and then performing an export.