The normal approximation is notoriously [1][2] bad when the binomial proportion you're trying to estimate is small and you don't have a big n. If one still wants a closed-form out-of-the-box formula (to avoid the exact resolution you suggested), one can use the Wilson approximation for instance.

[1]: http://en.wikipedia.org/wiki/Binomial_proportion_confidence_...

[2]: http://projecteuclid.org/download/pdf_1/euclid.ss/1009213286