I encountered the same problem and here is what I did. First, I downloaded and installed the Java2M Runtime Environment as indicated on the Download page:

**Users who download this version must already have the Java2TM Runtime Environment, v. 1.3.1 (JRE) installed. You can get the JRE from the Sun web site at: http://java.sun.com/j2se/1.3/jre/

Then I rebooted my computer and began to install LimeWire again and it installed just fine.