You have 1 ppm of ammonia right out of the tap? That's not good at all. 1 ppm of ammonia is lethal to fish, not instantly but pretty quickly. There are some ways you might be able to get around that problem but none are pretty.
The best is to get a RO/DI unit that will make pure water for you. Unfortunately you'll have to buy a supplement to add to the RO/DI water to replace vital minerals the RO/DI unit will remove and add some buffering capacity. Before you spend a lot of money on an RO/DI unit you might want to contact the water company and find out if they are adding ammonia and if they are if its temporary. In fact if you're getting those kind of readings you should do that regardless. Also have you tested for nitrites out of the tap?
There are some products that will change ammonia (toxic to fish) to ammonium (not toxic) but dosing would be something you'd have to do constantly and your ammonia test would always reveal the presence of ammonia just because it doesn't diferentiate between the two. Those products might let you get away without an RO/DI unit but it's iffy.
If you have to add fish right now it's not the end of the world but you need to be prepared for a lot of work and potentially losing some fish. If you go the detoxifier route Prime ( Aquarium Water Quality & Conditioners: Seachem Prime Water Conditioner
)has a good track record. According to the label it looks like you'd need to double dose in order to detoxify your ammonia levels. Just to be on the safe side I'd triple the dose.
Ok, so the steps.
1) Buy a product like Prime. Your LFS should stock Prime or something like it. Get a decent sized bottle, you're going to need it.
2) Do a water change. Call it 50% to get your nitrite levels down. Refill the tank.
3) Treat your entire tank with the Prime. Triple the dose.
4) Now acclimate and add your fish as normal.
5) When you do water changes you need to treat any water you add with that same triple dose of conditioner until your tap water no longer tests out as having ammonia in it.
Here's the disclaimer. I have no real experience with using a product like this to detoxify ammonia and nitrites. The science behind it is sound but I have no experience backing it up. You will still test as having ammonia in the water. You will want to carry out fish in cycling as normal but don't do water changes until ammonia levels rise above 1.25 ppm or nitrites rise above 0.25ppm. Keep a close eye on your fish. Once your filter matures it'll be able to remove the ammonia from the water and you'll see the level drop to 0. Be aware that even with a mature filter you still need to over dose with the conditioner during water changes.
All that being said, if the water company tells you the ammonia levels are going to be permanent then an RO/DI unit is your best bet. It'll remove the ammonia for you and you won't have to mess with overdosing Prime.
If there's anyway you can avoid picking up fish then I'd advise it and let your tank cycle without putting them through the stress.