This is quite similar to this question which asks if "types of fish" or "types of fishes" is correct. My conclusion was that both are acceptable.
In the case of birds too, it would seem that neither is incorrect. Although I'm not a huge fan of Ngrams, this one does at least show that both are steadily in use, although these is a clear bias towards the plural "species of birds". Looking at a sampling of the books using these terms though there appear to be scholarly works using either "bird" or "birds".
For example, the book "Conserving Bird Biodiversity" published by the Cambridge University Press and partially credited to the Zoological Society of London says:
In its latest assessment, BirdLife International describes 1,186 species of bird that are currently threatened with extinction worldwide.
And who are we to argue with the Zoological Society of London? :)