The handyman accused of raping and murdering Angelika Kluk left a fingerprint on tape used to gag her, a jury has heard.
Forensic scientist Carol Weston gave evidence at the High Court
An expert told the High Court in Edinburgh that the chances of the print coming from anyone else were about one in a billion.
The trial of Peter Tobin, 60, paused for two days last week after he was taken ill and was taken to hospital.
Mr Tobin has denied raping and murdering the Polish student.
The court heard about DNA evidence apparently linking the accused and the victim. It was suggested that Mr Tobin probably had sex with Angelika, 23, around the time of her death.
The jury was also shown photos of the blood-spattered garage attached to a Glasgow church where it is thought she was attacked.
Forensic scientist Carol Weston, 33, was called to St Patrick's Church in Anderston, Glasgow, after the body of the Polish student was found under the floor last September.
Mrs Weston said: "I was under the floor about, I think, for just under three hours."
She said the woman's bound and blood-stained body could be seen through a small hatch near the priest's confessional.
There was no blood in the area of the hatch but blood-stains were found in the church's garage - on the floor, almost 6ft up a concrete pillar and even on the ceiling.
DNA from the blood matched that of Angelika, the court heard.
Mrs Weston went on to say that semen found on Angelika's clothing was tested in Strathclyde Police's laboratory.
Advocate depute Dorothy Bain, prosecuting, said: "The sperm cells were found to match the DNA profile of Peter Tobin."
Mrs Weston said her examination suggested that Angelika had sex "near to the time of her death."
The student had not had time to move about or take a shower or a bath. She added that the probability of finding this match in another man unrelated to Mr Tobin was one in one billion.
The court heard that as the post-mortem examination on Angelika began, an insulating tape gag was removed and a kitchen cloth was taken from the dead student's mouth.
Forensic tests revealed a fingerprint on the tape, which was sampled for DNA.
Ms Bain asked: "The DNA taken by swabbing the fingerprint matched the DNA of who?"
"Peter Tobin," said Mrs Weston. The scientist said that again it was "a complete match."
She added: "The probability of the DNA profile originating from another male, unrelated to Peter Tobin, are estimated to be in the order of one in one billion."
Angelika's body was discovered under the floor of the church
Mrs Weston said the blood-stained kitchen cloth was examined and a single sperm cell was found.
It too matched Mr Tobin's profile, with a one in 210 million chance of it coming from another unrelated man.
The court also heard that microscopic blood-stains were found on the watch Mr Tobin was wearing when he turned up at a London hospital, three days after Angelika's body was found.
Mrs Weston said: "They were consistent with the watch being more heavily blood-stained and the blood being wiped off."
Further DNA evidence was found on Mr Tobin's T-shirt and the court also heard that a pair of blood-stained jeans were found in a wheelie bin at St Patrick's Church.
Mrs Weston said that when she examined the jeans the left knee was heavily blood-stained - as if someone had knelt in a pool of blood.
She said that there was a one in a billion chance that the blood came from someone other than Angelika.
The court heard cells found in the waistband of the jeans were tested for DNA and the profile matched that of Peter Tobin.
Mrs Weston confirmed that the more someone wore an article of clothing the more likely it was that DNA would be transferred.
Mr Tobin has denied murder, rape, attempting to defeat the ends of justice, attempting to pervert the course of justice and breach of the peace.
In a special defence read to the jury, Mr Tobin has admitted having sex with the Polish student, with her consent.
The trial continues.