How to go about a paternity test?

I had once with another person at the time I conceived. I cannot remember fully, but I'm almost 99% positive that there isn't the possibility that it could be this one persons. My now-husband and I must have had every night and maybe even twice a day for this month and some following it, unprotected. I hear that it is difficult for some people to conceive and I don't believe it would've been possible from this one other incident, but I have a twinge of doubt and do not like it. I am interested in getting tests done to verify that my now-husband is definitely the father. How would I go about doing this? How much would it cost out-of-pocket? Is there a way to do it at-home? Are these always effective or is there the possibility it could be wrong? Any advice on the matter would be very helpful, just no suggestions to go on Maury, please. Thanks!