I would start with the critical lists of the American Livestock Breeds Conservancy and the Society for the Perservation of Poultry Antiquities. Those organizations also have lists of member breeders that offer stock for sale. I don't know that it can be singled down to one specific breed however.
The ALBC is a good list. Then there are breeds out there which we have never even heard. Not all breesd are recognized or standardizes, which means there are probably thousands out there we've not seen or heard the names of.
Off the top of my head (based on my experience on rare chickens boards, talking to breeders, etc), I'd guess Chanteclers or Barred Hollands. I mean the real thing, not necessarily something the hatcheries are selling. So, Hollands that really lay white eggs (as the breed is known for), for instance, are nearly impossible to find.
According to ALBC it would be the Barred Holland, but I'm sure there is a breed not well known that is rarer. Also, if there are any unknown breeders of White Hollands they would be rarer than the Barred.
I don't think there are clear enough records kept of poultry numbers to be able to define the 'rarest' breed.
There are lots and lots of breeds that could benefit from additional breeders though. As the others stated, just take a peek at the American Livestock Breeds Conservancy to get an idea of just some of the breeds that could use attention.
Sometimes even if a breed itself is not rare, an approved color variation within a breed will be rare. I find White Rosecombs very hard to find, and I don't think many people breed Hamburgs in the other colors other than Silver Spangled.
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Case in point I recently found out that there are Ga Cua in the US. (Ga Cua are a Vietnemese game breed
I'm not allowed to link to the picture since it's on another poultry forum and that's against the rules but take my word they are absolutely stunning birds) I would be willing to bet that there aren't many in the US and those that are here are almost all closely held probably by the original importers or their descendants