Do you know what this connector is called?
Looks like a broken MMCX male connector to me. These are popular antenna connectors for embedded GPS receivers.


As the comments mention, it would be useful to have a photo of the mating socket to be sure.

Yes, you can purchase replacement connectors to be soldered + crimped onto the existing cable. But they're very fiddly. I would be inclined to either just buy a new antenna complete with connectorized pigtail, or cut the existing cable and crimp on an SMA plug to use with a short SMA to MMCX adapter cable, which are readily available.