They are treating the death as suspicious, but are waiting for the results of a post-mortem examination due today.
Miss Yeates, 25, from Clifton, Bristol, had not been seen since December 17 when CCTV showed her in a Tesco store on her way home to her flat in Canynge Road.
She lived with her boyfriend, Greg Reardon, 27, who reported her missing on the Sunday night after returning home from a weekend away in Sheffield visiting family.
Friends launched a campaign to find her by putting up posters across Bristol and setting up a website, a page on Facebook and using Twitter.
Tributes have since been paid to Miss Yeates – known as Jo – on the Facebook site.
Bec Wood wrote: "Rest In Peace Jo. You will always be loved and remembered for being so beautiful, kind, successful, and lovely. You made Greg so happy. Our thoughts are with Greg, your parents and all of your families and friends. We will miss you Jo, and hope that you have peace now. Love and prayers xxxxxx"
The body was found at 9am on Saturday on a snow-covered verge three miles from where Miss Yeates lived.
A couple had been walking their dogs along Longwood Lane, close to Bristol and Clifton Golf Club, which was later sealed off for forensic examination.
Police said the formal identification procedure was yet to be completed, but they spoke to Miss Yeates' family yesterday.
An Avon and Somerset police spokesman said: "Police are satisfied that the body is that of 25-year-old landscape architect Joanna Yeates, who went missing during the weekend before Christmas."
Miss Yeates' father said he "would be relieved" if the body was his daughter so they would have the opportunity to say goodbye.
David Yeates said: "It hasn't been a Christmas for us. I don't know what it's been. It's been surreal, totally unreal. In truth we feel like we hope ... we would be relieved if it was her.
"We don't want to go to our graves wondering where she is and
not being given the chance to say goodbye to her."
The body was clothed and covered in snow. Avon Fire Service helped police to remove from the scene to preserve the site for a thorough forensic examination.
Yesterday police continued their fingertip search in the Longwood Lane area and have said they would be looking at footage from the Clifton suspension bridge, which is the main route from Miss Yeates' house to where she was found.
Chief Superintendent Jon Stratford said: "Our heartfelt condolences go out to Joanna's family for their loss.
"We have not stopped working hard throughout the Christmas period to find their daughter after she was reported missing.
"Now we will work just as hard to discover exactly what happened to her and how she came to be in Longwood Lane on Christmas morning.
"Until the post-mortem examination is able to firmly establish how Joanna died, we are keeping an open mind about the cause of her death.
"However, I would appeal to anyone with any information whatsoever that can help this investigation to please come forward and help us provide Joanna's parents with the answers they so desperately want and need."
A post-mortem examination took place in Bristol, but due to the freezing conditions in which Miss Yeates' body was found, results are not expected until today at the earliest.
We don't want to go to our graves wondering where she is and not being given the chance to say goodbye to her.