Following a long-running battle with the community and local Green Party MP Caroline Lucas, Tesco community and government director David North confirmed that the supermarket was no longer looking to open on the site.
"The future of the community garden should be resolved locally, by the community and its representatives," he said.
The former petrol station site on the outskirts of the city was being used as a garden for local residents with the permission of the owners.
Upon the announcement that Tesco was looking to buy the site and set up a store, residents and local shops campaigned against the supermarket chain, fending off bulldozers on several occasions.
Lucas said she was pleased by Tesco's U-turn. "I pay tribute to all the local people who have fought a tireless campaign to preserve the community garden, and I am delighted that Tesco has listened to the concerns the local community."