Take the Zexonyte to Dr. Andonuts in Saturn Valley. He'll get to work on the Phase Distorter II. Rest in the hotel, and the Phase Distorter II will be done the next day. Check out the new items in the Saturn Valley store. Then get yourself all stocked up and ready, because after you get into the Phase Distorter II, you won't be able to go back.
If you're ready, and you're all stocked for a difficult dungeon and final battle, get into the Phase Distorter II.
You'll end up in the cave that you might have briefly visited in the Lost Underworld, with the weird, creepy music, the broken machine, and the metallic tentacle. That broken machine is the Phase Distorter that Pokey stole.
As you head east from where the Phase Distorter II lands, the Star Master will swoop down and grant Poo the power of PSI Starstorm Ω. Then you'll reach the broken Phase Distorter and the formerly kidnapped Mr. Saturn. Check the broken Phase Distorter for a Horn of Life.
Now, just wait around. Soon, a Phase Distorter III will appear out of nowhere. Apparently, this machine can travel not only through space, but through time as well. Dr. Andonuts and the others didn't use the time travel feature to get here, though.
Dr. Andonuts has some difficult news for you. You need to travel to the distant past to fight Giygas, but when the Phase Distorter III travels through time, all organic matter inside gets destroyed in the process. So in order for Ness and the others to safely travel to the past, you all have to be transferred to robot bodies. That is really really creepy. At least I think so.
Anyway, you have no choice but to watch as Dr. Andonuts transfers you all into robot bodies. He gives Ness his baseball cap, but the others all look pretty generic. And then, you're sent off to the past.