At a press conference at their home, Mr Hawker praised Japanese police who, he said, "after a very bungled start" to the inquiry "had got their man".
He said the family was relieved and added: "I can go down to my daughter's grave this afternoon to tell her."
Mr Hawker said he wanted the Japanese authorities to punish Ichihashi "severely".
"We accept whatever the Japanese authorities have to do to him, if that is life imprisonment or if he gets capital punishment."
Earlier Mr Ichihashi's mother appeared on television urging her son to turn himself in to police.
A Foreign Office spokeswoman said: "We are glad and relieved that the only suspect in this case, Tatsuya Ichihashi, has been found and is now in police custody.
"We are grateful to the Japanese police for their efforts and we remain in close contact with the Hawker family."
A spokesman for the Osaka Prefectural Police said: "We have detained a person in Osaka city who appears to be the suspect in the murder."
Bill Hawker: "We were always confident the Japanese police would eventually find him"
Miss Hawker, a Leeds University graduate, was teaching English at Koiwa school in Tokyo when she went missing on 25 March, 2007.
Her body was discovered on the balcony of Mr Ichihashi's flat, taped-up and half-buried in a bathtub. The suspect escaped as police arrived at his flat.
According to unconfirmed reports in the Japanese media, police detained a man believed to be Mr Ichihashi at Osaka ferry port at about 1800 local time.
It was suggested he was waiting for a ferry bound for the island of Okinawa in the far south-west of Japan.
Incident 'shame'
He is reported to have told officers: "I am Ichihashi."
Mr Ichihashi is thought to have recently had plastic surgery to change his appearance. He is believed to have had cheek implants and lip-thinning.
Police went to the clinic where a plastic surgery patient believed to be Mr Ichihashi had a follow-up appointment, but he never arrived.
It was reported that detectives then examined photographs provided by the hospital and believed their patient was the suspect.
Miss Hawker's family had visited Japan several times, appealing to police and the public for help in finding the suspect.
Her father appealed to the Japanese public for her killer to be caught and said the incident "brought shame" on the country.
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