personally I'd stretch your budget a little bit more to get something like this:http://www.newegg.com/Product/Product.aspx?Item=N82E16814161355&cm_re=6950-_-14-161-355-_-Product
or if you really can't go above budget the 1 gig version is fine as well.
a 6870 will serve its purpose and I have ran a single 6870 with an oc and it does fine at 1080p, but I think it would be worth the investment to bump up to the 6950 2 gig if possible, it should last you a bit longer anyways, another option might be the gtx 560ti if you can find a good deal on one. Overall I would stretch to a 6950 2gig if you can, it will make you investment last a bit longer and really its only like 20 or so more than $230, that would be the smart move imo.
and also like maziar said make sure you have ample psu. If you are unsure, what is your psu make and model?