Are you downloading LW using the offline installer?

First of all, download Java. Use the JRE offline installer & remove your current Java version before you install the new one.
Java SE Downloads

Now delete LW's preferences folder. See point 3 here (also point 2 if you've used much older versions of LW in the past).
Fixes for LimeWire

Last of all, reinstall LW & make sure to use the offline installer (not the recommended one).
LimeWire: The Official Site for the Fastest File Sharing Program on the Planet

If none of that helps, try the 'other' installer here.
Installing using the 'other' installer