Just to be sure about the Zithromax concentration, you said that the box states 200mg/5 ml (that would be the conc'n when all the powder is mixed with the water). You also said that the vet had suggested mixing 1/8 of the powder with 9 ml water. If you did that, then won't you have a much weaker solution?
With a small rat (yours is 1/3 lb) you would use .04 ml of the usual concentration at the low end of the recommended dosage range(Debbie D's book recommends 5-15 mg/lb, and my vet usually goes with the low end of the range). If you have diluted the mixture with only 1/8 of the powder then you would need to give a higher volume, .32ml. You might have trouble getting her to take that amount. My rats find it bitter, so I have to mix the meds with lots of stuff to disguise the taste.
Sorry if this is confusing. If you are really just making up a smaller amount of the full concentration, then your vet's suggestion of .06 ml is great (the mid-range dose).
Sounds like you now have a good vet.