We derive theoretical upper and lower bounds on the maximum size of DNA codes of length $n$ with constant GC-content $w$ and minimum Hamming distance $d$, both with and without the additional constraint that the minimum Hamming distance between any codeword and the reverse-complement of any codeword be at least $d$. We also explicitly construct codes that are larger than the best previously-published codes for many choices of the parameters $n$, $d$ and $w$.