CELESTION SL6 Si - woofer replacement/repair

Give Bill Legall from Millersound a call to get his opinion on this situation;he is well known in the business and highly respected for his work.

Millersound
1422 Taylor Road
Lansdale, PA 19446-1531
(215) 412-7700

If you have owned the SL series of Celestion speakers, you will know that these can't be reconed or have the surrounds replaced. The driver was proprietary to Celestion. It was unique in not having a dust cap (which was unusual at that time), and changing it for a different driver would no doubt change the sound of the speaker completely.
Elizabeth's comment is a good one. These drivers need to be checked for tightness on a regular basis. Also check the ribbed aluminum diffraction plates around the drivers for tightness.
You didn't say if you had found any cracks in the surround. If you have, and that turns out to be what is causing the buzzing that you hear, I think that the only alternative will be finding a used driver.

thanks for all the great feedback. what should i do (if anything) to determine what the source of the buzzing is? - i don't want to do anything to make things worse if i can help it, obviously. this may not be a good thing to do, but i very very gently applied a feather-light amount of pressure to the surround while it was playing and the buzzing went away - not sure what that might indicate.

"not sure what that might indicate"

It means the woofer is beat. Like Rleff said, get a hold of Millersound and have them fix the woofer.. They may be the best speaker repair shop in the country.