There is another stage where you can obtain this achievement, which is IMO quite a bit easier and safer from accidental premature mission ending leading to thrown xbox controller into tv screen. No one wants that do they?
So this is the lowdown: use the stage "Chatéau Notre Desir - Uranus." The prime reason being that it's very easy to accidently roll up the black hole and end the stage early before you're ready on Schloss Kosmos where you are additionally burdened with a time limit, albeit a generous one, but for this run you will want all the time you can get to leasurely pick up everything from smallest to largest with no hurry at all to approach large size once in space.
On "Chatéau Notre Desir - Uranus" this method ONLY works if your memory score for the run where you make a 1.5million km katamari is your new high score for memories on that stage as well. The key to getting lots of memories is staying as small as possible avoiding everything you can that is not requested by the king. The size of the item is mostly unrelated to it's memory value. IE: a diamond ring is worth about the same a limo if the king requests "luxury" for example. The reason for this is once he reaches the size where you can roll up large houses he will immediately stop giving requests meaning you can't get any more memories on that run, but until he gets that big, he will give a potentially unlimited number of requests. On my first try just trying that strategy (staying as small as possible) to see if it would work I easily got 2600 memories. If you run the stage and dont manage to beat your memory highscore before the king stops giving requests, just restart, because it won't register unless it's the record highest memory run you've done on that stage because it's linked to the primary mission criteria, I suppose.
Again, what makes "Chatéau Notre Desir - Uranus" easier than Schloss is that you can roll up the black hole without automatically ending the stage and there is no time limit- as the king says: "just keep rolling till you're done and hit x. "
Groudon is correct in that you will need to complete stages multiple times because that spawns more planets out in space (that you created playing the other stages) to add to your size.
Some stages can only turn out very small planets, but Elem3ntal80 lays it out in the easiest manner possible:
"- The larger the katamari is, the more size it will be worth when you roll it up in space, so you want to especially play the levels several times where you create larger katamari. For this reason, the Dangerous and Calorie DLC stages are the best if you don't want to get them all. You can have a max of 10 satellites per level which you can see by going into the Cosmos while flying around in Princedom."
So that means you can make TEN suns and TEN Cetus' which produce massive planets/suns which are worth quite a bit of size. Lastly, try to roll up everything as small as you can in space avoiding picking up planets etc until you're almost completely running out of tiny regular stars to pick up, because if you roll up the queen or something large accidently you'll go up quite a bit in size making those stars too small to appear and unable to be rolled up.
Good luck... but you won't need it if you just follow the advice compiled here! :)