I had already looked into this, so I can readily tell you that the building in question was built as a department store at the turn of the century. It's 12 stories tall, with the street level occupied by a couple retail stores.
According to Amazon's new landlord, Vornado Realty Trust, each floor has around 43,000 sq. ft of space to lease. They also said that Amazon is renting 470,000 square feet, which is actually more floor space than the entire building contains.
In short, it looks like this is going to be Amazon's new New York City office building. Whether it will also be Amazon's first retail store is another matter.
I don't know anything about Amazon's office space in NYC, but I would expect that they at least lease a floor here and there. But with the lease of the entire building at 7 West 34th Street, it is very likely that this will become Amazon's main office in New York City (assuming they don't have one already).
But will this also be a store? We'll just have to wait and see.
Update: The WSJ reports that:
Amazon is planning to convert some of the space into a mini-warehouse, with limited inventory for same-day delivery within New York, product returns and exchanges, and pickups of online orders, according to people familiar with the matter.
Over time, Amazon could use the space to showcase some inventory, such as its Kindle e-readers, Fire tablets and set-top box, among other merchandise, these people said.
Amazon's current physical retail efforts are limited to less than permanent pop up stores (including a couple in California this holiday season), delivery lockers, and vending machines installed in airports and other high traffic locations.
Update: Amazon told Geekwire that: "we have leased this building primarily as corporate office space and we intend to sublease to other tenants the ground floor retail space".
There's been a lot of talk about Amazon opening a store, but so far that's all it's been: talk, and almost none of it is coming from Amazon.