Congrats on the babies!!!
I have several guppy
fry in a 1 gallon tank in my daughter's room at the moment. It does have a (crappy) under-gravel filter and a heater that rarely goes on (this tank is more likely to get overheated when the lights are on than to get too chill for the fish - either way, keep a close eye on the temperature fluctuations)
It's possible to do, but it takes a LOT of work to keep the water parameters stable in such a tiny tank. A cycle can be established, though it takes some time. . . and you'll need to do insane amounts of water changes in a tank this small to keep the water clean enough, and there are several problems that come up here.
You must feed fry frequently with small amounts of high-protein food. This creates more than your normal amount of waste in the tank, which also produces a rapid rise in ammonia and other toxins - which means you have to do MORE water changes to keep it clean.
In order to keep such a tiny body of water clear of toxins, you have to vacuum (if you have gravel) to get all the microscopic bits of bitty fish food and poo out. Vacuuming a tank this small is no easy feat. Even if you don't have gravel - siphons and fry do NOT a good combo make - especially in close quarters!
Fry have tiny, super delicate systems that are not fully matured yet. . . in any tank, if your water parameters are off fry will be the first to go - their tender little gills simply can not handle toxins - so you HAVE to constantly check and change the water - and that's even in my 1g that's cycled already!
ONE GALLON is NOT a lot of water. It's stressful on the fish to drain the tank too low and refill it, also stressful on them if you were to try moving them back and forth every time you (constantly) do a water change. . .so it's best to do it in 2-4 25% water changes. . . very gently - they're tiny babies in there!!!
In my tiny tank, I have 10 fry. GUPPY fry. . . I also own Mollies, and their babies are far larger from the start than guppies. Plus Mollies are known for being messy fish - they have a bioload that's larger than you'd think. I do not think I
would be comfortable trying to keep Molly fry in a tank this small. Another thought for you is this. . . when I had my last batch of Molly fry, I separated some and put them into a breeder basket in a 10
gallon tank. After I did, my female had 5 more fry, which I left in the 29g tank where they were born, just to see. The 5 survived just fine in the big tank, and after a while I noticed that the fry in the larger tank grew more quickly and were stronger than the ones in the smaller 10g breeder box. I've noted the same thing with the guppies in the 1g vs those that have been around in the 10g. I've read that fish may release a certain hormone to regulate their growth rate. It's very possible that having the fry in such close quarters for too long could stunt their growth - at the very least, don't be surprised if they don't grow as quickly as you'd expect. After a few weeks I moved the Mollies that had been in the 10g back into the 29 - it's several months later, and I can still EASILY pick out the five (much bigger and stronger) fry that stayed in the 29.
Congrats on your babies, though! I don't know if you were asking for advice, but I figured I'd share since I've just been through it myself. I agree with the PP - it would be best to get one of the 5g kits, they have them on sale for $20ish very frequently. Take good care of that water and those adorable little Molly babes. I just love Mollies! :)