Because of "overwhelming demand" for the hot new device, Sprint has had to push back the ship dates on most of their Samsung Galaxy S III 16GB orders. Additionally, insufficient supply also slowed things down. Because of that, they pretty much missed their target date of June 21st for offering the phone in the retail chain, although they were able to ship the previous pre-orders out. Also, just 1 day ago they were able to ship out pre-orders for the 32GB version. Since then, the pre-order page has simply stated, "Sorry, we are out-of-stock. Pre-Order Closed. Check back."

Now we have new word from Sprint indicating that the phone will be coming on July 1st, although it is still possible that even that date could get pushed back. Here's a quote with some details,

The last piece of the puzzle came today, in the form of an official announcement on Sprint Community, stating that the Samsung Galaxy S3 “will be available at Sprint beginning on Sunday, July 1.” According to one of the forum’s admins, the 16 GB version should be up for sale in three days from Sprint Stores, Sprint Business Sales, Telesales and Web Sales, while the 32 GB model will only be available from Web Sales, Sprint Business Sales, and Telesales for starters.
Sound off in the forums if you get yours on Sunday or a few days thereafter, and we will keep you informed in case something goes wrong. It's good to see that Sprint is bending over backwards to try and get these orders filled.

Source: S3Forums via AndroidAuthority