In the western part of the United States, the term "currant" usually refers to dried fruit from the gooseberry plant. Quite expensive now. I checked with Merriam-Webster Online and found the following definition. They call the Ribes (as opposed to Vitus) type a "red currant":

"One entry found for red currant.

Main Entry:        red currant
Function:        noun
: either of two currants (Ribes sativum and R. rubrum) often cultivated for their fruit; also : the fruit"


Out of curiosity, I checked what they had to say about the Zante grape and the Zante currant. Nothing available, but their suggestions for an alternate search seem a bit humorous:

Suggestions for zante grape:

         1. centigrade
         2. centigram
         3. centaureas
         4. centered
         5. Sainte-Therese
         6. centigrams
         7. scene-stealer
         8. saunterer
         9. Szent-Gyorgyi
        10. Saint-Germain