From my own perspective, after seeing the talent and effort being poured into the Linux development process, Weiss' prediction looks like a safe bet, even if the backers of HP UX, IBM's AIX, and Solaris don't see things that way. New applications will be written for Linux rather than Unix by the end 2009 because it will do 95% of the things that Unix does, and the kernel development process will be closing out the remaining differences fast.
Unix will maintain an edge in scalability until then, then Linux will catch up. Linux will excel at running on multicore chips and multiprocessor servers and at implementing virtualization. Unix has some advanced identity management and security features available to run on it, but Linux will catch up. Solaris leads in terms of system self-diagnostic capabilities, but Linux will catch up.
That doesn't mean Unix stops running sometime after 2009. On the contrary, Unix will be in operation many years from now. New applications developed for Linux will, in some cases, be easily converted to run under Unix. Weiss didn't say no new applications will run on Unix; he said new applications won't be developed with Unix as their primary target system. Those days are over.
Linux has its roots in Unix and is POSIX (Portable Operating System Interface) compliant, so converting a new application from one to the other will always be possible. But what Weiss' comment really means is that there are now two major operating systems with the resources behind them to stay in the forefront of modern systems. One is Windows Server, with Microsoft's millions being poured into it. The other is Linux, with the Linux kernel development process.