Family Records Centre (main office in Myddleton St., Finsbury, as already posted) should be your main starting point. There is a link here:
http://www.familyrecords.gov.uk/topics/bmd_3.h tm#engwales
You can apply on-line.
However, for births more recent than 50 years ago you will need to provide ID. The problem is that birth certs are highly useful in identity theft or similar fraud and even if you only wanted to know the birthdate, your enquiry would, in these highly cautious times, be viewed with suspicion. Birthdates are key information for obtaining info on someone.
However assuming your enquiry is completely innocent (it's always nice to think the best of people), no-one has yet suggested that the birth might have been announced in national papers (here you would probably need
The Times or the
Telegraph). The
Times is available in big libraries (usually on microfilm) but unfortunately the
Telegraph is far more popular for birth announcements. Some big libraries may have it, otherwise it's a trip to Colindale (London NW9) for the National Newspaper Library (part of the British Library).
Local papers in London (if that's where you think the birth took place) are not so useful unless you know the part of London, since local papers are truly local, e.g. the
South London Press or the
Hampstead and Highgate Express.