Best Record Store In New York

I will be visiting New York for the first time this coming 12/22/09 and am interested in going to a record store that has a large selection of Jazz and Classical. Any recommendations? Thanks for your help!
If you find yourself in Philadelphia, try Philadelphia Record Exchange on S. 5th St. They have a very good selection of used vinyl at reasonable prices in all categories of music. It isn't NYC, but it's close enough. The last time I was in Academy Records, the selection was next to zero. Princeton Record Exchange has a pretty good selection of jazz and rock and I was there just 6 weeks ago. Of course, all of these places have a varying amount of vinyl as they buy and sell. It's the luck of the draw and usually requires several visits in order to come up with really great finds. Used vinyl stores are often cherry picked by on-line dealers who sell on ebay or on their own websites at premium prices.
This is not a recommendation for everyone but if your jazz tastes run to free jazz, improvisational jazz, and modern jazz, as well as modern classical, and the late 60s and early 70s progressive scene, there is one place that is absolutely the best place on the planet on a per square foot basis (because it is a small basement shop): The Downtown Music Gallery (now in Chinatown). Check it out:

Think Andrew Hill, Cecil Taylor, John Zorn, Keith Tippett, Anthony Braxton, Soft Machine, Terry Riley, Phillip Glass, King Crimson, X-Legged Sally, Miles Davis, Coltrane, Ornette Coleman, Brotherhood of Breath, Robert Wyatt, Last Exit, Peter Brotzmann, Mathew Shipp, William Parker, Larval, Gunter Hampel, Misha Mengelberg,..."that" sort of thing.

You've never seen most of the stuff nor heard most of the artists, but they've got stuff playing all the time in there and I've picked up many things just because I was there and heard something playing. I've spent a couple of hours in there on more than one occasion. A real treasure.
Thanks for all the info! All of the places above look great! I will be in New York for a few days, so I should be able to visit most of these shops! Thanks again for all the recommendations!
I remember years ago when I was with CBS, there was a record store in the Village area of New York. But I can't recall the name of the store or if it is still there. Do remember always had a great selection and prices were very friendly. Scooped a lot of great vinyl there.