Verizon has opened up pre-orders for the Galaxy S4 on their website today. Pricing is $249.99 with a $50 mail-in rebate debit card (why not just give us the rebate without the hassle of the card?). A 2 year contract is required, so with the rebate, the phone is effectively $199.99. Otherwise the phone is $649.99 to purchase without a contract.
The page also confirms that the ship date is May 30th, or at least, it states the phone will ship by 5/30.
So, there you have it. The mystery has been solved. We now know the cost and a final confirmation of when the phone should show up on the Verizon Wireless network.