Set Java 6 as the default via the Java control panel. This solved the issue for me on Windows 7. In that way you can have both versions installed.

At this point, Java 7 has not yet become the default Java to download from the Java site. I do wonder why and whether it is fully developed.
Why LimeWire will not run with it I do not know, one guess is the program was only set to use Java 6 and not 6+, a silly oversight. Presumably a mistake by the original LW 5 developers.

I will ask File_Girl71 if she can possibly check if this is the case with a simple + symbol missing, and redo a LPE version.