Having the 6-pin connector is not going to make a difference in whether you can run it or not. Running a 6850 on 300W is taking a risk, and a big one. I seriously recommend a power supply upgrade to support the new graphics card you have purchased.
What kind of computer do you have?
On max load, the 6850 uses 125W. That doesnt leave much left on the 12V rail for your other things, namely the processor and such.
You can calculate how much power your system needshttp://www.newegg.com/Product/Product.aspx?Item=N82E16817371029
^That should work well for you, very highly rated.