Police said the grandfather, who they also identified as the mother's lover, admitted killing the child, Rose, and dumping her body in a river. However, he also gave several other accounts, including that he trafficked the girl to Palestinians, police said.
Ilan Ben-Shalom, head of the police team investigating Rose's disappearance from the Mediterranean coastal resort of Netanya, said the mother was arrested two weeks ago with her lover, who is the father of her former husband.
The arrests followed a tip from the woman's mother, who said she had not seen the child for several months and feared for her well-being, he said.
The partial lifting of a court-imposed gag order revealed a bizarre love triangle in which Rose's grandfather became her stepfather, falling in love with his son's young bride and fathering two more daughters with her.
"According to the suspects' statements so far, there is a strong suspicion that Roni and Marie are involved in violent abuse and the murder of Rose," Ben-Shalom said at a news conference, using only the first names of the two suspects.
But local media identified the grandfather as Roni Ron, a 45-year-old Israeli, and the mother as a Frenchwoman, Marie Pisam, 23.
Ron's lawyer, Revital Swid, said her client admitted hitting Rose in a burst of anger as she cried in the back seat of the car and that later, stricken by fear and remorse, he concealed her corpse. Ron said Marie was unaware of the killing.
"He confessed to investigators and reconstructed the events of that tragic night when he gave the child a blow, a blow he did not expect to have such a result," Swid told reporters. "He was under pressure, panicked, was very worried about his partner's likely reaction, and as a result took action to conceal (the body)."
Israel's national police commissioner, Dudi Cohen, said Ron have given several conflicting statements. He said police were focusing on his latest version in which he said he took a suitcase containing the girl's body to the Yarkin river, about 15 miles south of Netanya.
No body has been found and there is no forensic evidence proving she is dead.
The case has absorbed the Israeli media since police launched a public appeal Sunday for information on Rose's whereabouts. The blue-eyed, brown-haired girl's picture filled the front pages of newspapers and was featured on TV and radio newscasts, eclipsing a visit by the U.S. secretary of state and the conflict with the Palestinians.
"Losing hope" was the banner headline in the mass-circulation Yediot Ahronot, along with a picture of Rose in a pink dress, gazing into the camera from under auburn ringlets. "A lousy day to be a police detective," Haaretz said over its front-page story.
However, all mention of a murder investigation was barred by the court order until Tuesday afternoon.
Once the floodgates opened, a picture began to emerge of a tragic child from a deeply dysfunctional family.
Speaking on condition of anonymity because of the remaining reporting restrictions, police investigators gave this account:
Rose's father, Benjamin, was the illegitimate offspring of a brief relationship between Roni Ron and a French tourist and the boy grew up in France without knowing his own father, police said.
Benjamin met Marie in France, and after Rose's birth the teenagers got married and visited Israel to explore Benjamin's roots. After meeting Ron, however, Marie fell in love with him and stayed on in Israel after Benjamin and Rose returned to France.
A child custody battle ensued, amid suspicions of abuse of Rose by her father. A French court awarded custody to Marie and she came back to Israel with the child, setting up a home with Ron and giving birth to two more daughters, now 1 and 2 years old.
Netanya's welfare department said the two young girls were now in the care of social services.
Benjamin, the missing child's father, is in France.
"At this stage he is refusing to cooperate with the Israeli police, and we cannot legally force him to do so," Cohen told reporters.