Honestly, if I tested the water straight out of my tap and got a reading of 1ppm of ammonia, I'd switch to R/O water with a mineralizer like Seachem Equilibrium. Prime will help detoxify the ammonia, but personally, I think it's intended to help with much smaller concentrations than 1ppm.
Plants and a cycled tank are definitely helpful, but every water change you do is adding so much ammonia that the nitrates are going to zip right back up--what a nightmare. Since the tank is relatively small anyway, it wouldn't be very expensive to do water changes with R/O water--most pet stores offer it at a very reasonable price. My LFS carges $0.75 for five gallons, I think.
I hope you find a solution.