Hey the 6570 you own is not powerful enough to play BF3. You should mention where you stay as shipping is included in your budget.
Consider selling your 6570 or replacing it . It will give you more money to spend a little extra, as the budget is of constraint.
Also ensure you have a decent PSU to support these components.
Since you will be buying these new components consider selling the old motherboard,CPU RAM to fetch some more money.
Assuming you can cough $30 more, do look into this suggestion:
Will the motherboard fit my fan and the current things I have?
Most pre-assembled computers like the acer one will give problem in terms of upgrading. It's better if you can hold for some time until you get more money so that you can get a case and a decent power supply, like around $40-$50 for both. Get HDD , Optical Drive from the Acer desktop, and you will have a new system.