If JSSE doesn't work out for you (from @skaffman's answer, it may be hard to find documentation), you may want to look into some sort of a proxy. You could set up a daemon running on the same local machine (or trusted network), which then forwards the requests over HTTPS to the final end point. You could write this proxy server using a more modern JVM. Your legacy system would then point to the proxy rather than the real service.
Of course, if, by chance, you also have control over the final end point, you could perhaps just put both servers on a VPN.