Linking this library statically or dynamically with other modules is making a combined work based on this library. Thus, the terms and conditions of the GNU General Public License cover the whole combination.
As a special exception, the copyright holders of this library give you permission to link this library with independent modules to produce an executable, regardless of the license terms of these independent modules, and to copy and distribute the resulting executable under terms of your choice, provided that you also meet, for each linked independent module, the terms and conditions of the license of that module. An independent module is a module which is not derived from or based on this library. If you modify this library, you may extend this exception to your version of the library, but you are not obligated to do so. If you do not wish to do so, delete this exception statement from your version.
There's no distinction about static vs. dynamic linking, and no specific requirement like the LGPL to be able to relink the application (though the GPL's general requirement to produce useful source + build scripts for the library still applies).
I worked a bit on Classpath but that was a long time ago. There are like, lots and lots of emails + irc logs to dig up about that little exception and what it means. Not having things tested in court is a problem, and the lack of funding for FSF lawyers is another.
You can link it with whatever proprietary code you want, but keep the LGPL library free. As long as you don't want to link LGPL-licensed library statically, it's quite simple to comply with license terms.
There's also MPL (v2), that's weaker than LGPL, roughly operates on file-level (rather than LGPL's library-level).