Yes!

I was gonna do this for my birthday but did summat else instead... Their website (http://www.english-heritage.org.uk/stonehenge/) states:

"There is no access to the centre circle during normal opening hours, however, Stone Circle Access can be arranged out of hours by telephoning (01980) 626267."

It's £12 per person (!) or £10 if you're a member. They allow 26 people at any time, and if it's not full, you can book the day before. You need to pay up front, and can email the forms and pay over the phone.

I'm sure others will tell you to go on a Solstice, when it's free, but there's yer options! :o)

G x