The former Papa's Fish & Chip restaurant was demolished in Eastfield to make way for the new site which will also see a KFC restaurant built.
Developers Rothstone Estates completed the site purchase after securing planning permission for the development in December last year.
An interior fit-out of the new Costa shop is now well underway, with a planned opening in August.
The new site at Eastfield will also feature a KFC restaurant opposite Costa Coffee, with construction of the KFC drive-thru to begin later this summer.
A total of 24 new jobs will be created as a result of the development.
Mark Rothery, Managing Director at Rothstone Estates said: "The development will see the site being brought back into beneficial use following Papa’s decision to close the site to concentrate on its seafront restaurant.
"Work on-site continues at pace with an opening of the Costa next month and construction of the KFC beginning soon."
During the planning process concerns were raised about an increase in traffic and litter at the new coffee shop and restaurant as it will join the Market Lane McDonalds and Morrisons site in Eastfield; they were later approved.