The June 1st show at the Shrine Auditorium is now listed at Ticketmaster. Tickets go on sale at 11am pacific on April 19th. Prices are $42 - $82. Show starts at 7pm.
Update: Pre-sale will start on friday at 10am pacific. Password is HEAVEN.
Also, you can win tickets from KROQ:
"WIN TICKETS TO SEE THE CURE AT THE SHRINE! Tickets to The Cure at The Shrine, June 1st go on sale this Saturday at 11AM. You can win them before they go on sale by listening to KROQ of the 90's at noon with Kat. Be caller 20 to 800-520-1067 when she tells you to call."