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As per the title, current cheapest is £532 with Iberia with a stop or £697 with BA direct.
Try KAYAK. Don't promise it'll be cheaper but its an alternative.
+1 for Kayak. Just booked 12 flights and this found the cheapest each time. Lastminute.com is also pretty good - try phoning them if you have the time as the people there (I found) are really helpful and really do try to find the cheapest flights.
edit/ are you doing a simple return too? As booking a multi city flight, ie if you chose the stop over option, pick another route back as this often works out cheaper than doing the exact same route back.
I was going to fly in and out of rio or sao paulo.
Dates?
Kayak says £494 with Iberia with a quick March search.
Again if you chose the stop over option, try playing around with the cities you stop in as it sometimes makes it cheaper + if you are going to have to stop, might as well stop at a diff airport!
9th to 23rd, I was trying to avoid a 25 hour plus flight. I'll have a look, thanks v83
There was some decent prices flying from Lisbon
If you're going to SP and want to go direct from the UK, you really only have a choice between BA and Tam (the Brazilian carrier).
Tam is substantially cheaper, and frankly, there ain't much in it service-wise. Tam staff are all Brazilian though, so a bit of 'please and thank-you' type Portuguese doesn't go amiss when dealing with them. BA aircrew will all be British.
Over the last year, I've been about 10 times and went Tam for nearly all. Only went BA when I had late notice and couldn't get an economy Tam seat.
Economy on Tam isn't bad, but they charge extra for exit-row seats, and you can only book them at check-in, which means getting there early. Biz-class is lovely, but properly pricey.
As there's only one flight a day from Heathrow to SP on BA or Tam, the flight is always full, so never any chance of an upgrade even if you have the points for it.
Try Trailfinders
When looking for cheap flights, make sure you clear your cookies before returning to a site you've checked previously. That way they'll think you're a first time browser and won't put up the prices they show.
If you can find a cheap flight to Geneva ......
http://www.holidaypirates.com/others/flights-switzerland-rio-de-janeiro-451-trip-incl-taxes