Update: Looks dead as of 5:00 PM Eastern. Hope you got in!
This morning, The Flight Deal posted a cheap fare to Rio during June. The cheapest ($640) only works for specific dates, but I was messing around on the Cheapo Air website and was consistently finding flights for less than $1200 round trip all throughout the month of June. My guess is most people are going to pay $1600-$1800. You can only book these cheaper rates through Cheapo Air, and I only had time to check JFK-GIG on alternate dates from The Flight Deal’s original post. Personally, I think $1200 is a good rate – I’ve attached a screenshot of an example fare. Note it’s unclear whether you will earn miles on this – I’d book this solely to save money if you’re going to the World Cup. Even if you don’t live in NYC, you would probably save money overall if you bought a separate flight to JFK.
For example, if you book the $1100 flight above, and book a $300 domestic RT, your total ticket cost to Brazil is $1400 round trip. I think that’s a good price any summer, let alone during the World Cup. Cheapo Air only has a 4 hour cancellation policy if you change your mind, and you have to call, so bear all that in mind. Good luck! And again, hat tip to The Flight Deal! Great site that I recommend RSSing.