Apple has extended its service coverage for Apple Watch (1st gen) models experiencing expanded/swollen battery problems to three years, covering customers for service for an additional two years beyond the complimentary 1-year Limited Warranty that it provides with the device.
https://9to5mac.com/2017/04/28/apple-watch-expanded-swollen-battery-repair-coverage-warranty/
They will fix the watch for free. You need to call Apple; and provide this information about the extended coverage for series 1 only.