This is a procedure where the gynae would first do an ultrasound to locate the position of your baby, before inserting in a needle thru your tummy and withdrawing a little amniotic fluid to send for testing.
Not sure how much is this going to cost, but as this procedure carries a slight risk in miscarriage, it would not be advisable unless its really really important and necessary to have the test done.
Else, you can just hold on till you give birth to your precious baby, and then do a DNA test AFTER your baby is borne.
There is this commerical DNA lab, that does paternity testings. You might want to browse thru their webbie
i-DNA Genetics : DNA Testing, Research Laboratory Products & Services
or go to google for this company: i-DNA Genetics
From their website, paternity testing takes about 2 weeks and have a range of cost starting from $500.
Hope the above information would help you a little in making your decision.
The pregnancy, planned or not, is still ultimately YOUR baby.
Once you give birth to your bundle of joy, you can then reconsider if you really need the paternity test to be done.
All the best in your pregnancy journey and remember to take good care of yourself!