04/10/2007 at 01:10 PM EDT
The results of DNA tests that could determine whether Larry Birkhead is the father of Anna Nicole Smith's 7-month-old daughter, Dannielynn, will likely be announced on Tuesday.
The results are widely expected to be revealed at a Supreme Court hearing, but Bahamian law prohibits lawyers and officials from discussing the case or even revealing what the hearing is about.
Smith's companion, Howard K. Stern, 38, currently has custody of Dannielynn and is listed as her father on her Bahamian birth certificate. However, Birkhead, 34, a former boyfriend of Smith's, also claims he is the child's father. Only Birkhead submitted his DNA for comparison to Dannielynn's.
Dannielynn's DNA was tested on March 21. Stern had tried to block the release of the results, but last week dropped his appeal
Birkhead has been claiming paternity of Dannielynn since shortly after she was born in the Bahamas on Sept. 7 – three days before Smith's 20-year-old son, Daniel, died while visiting his mother in a Nassau hospital.
He told Access Hollywood
in February that while he was dating Smith, she told him, "Yes, we're having a baby."
Dannielynn, whose full name is Dannielynn Hope Marshall Stern, stands to inherit millions from the estate of Smith's late husband, Texas oil baron J. Howard Marshall II, who died in 1995. Smith, who died Feb. 8
, had been battling his family over the estimated $500 million fortune.
Once paternity is determined, the court, which must act in the best interests of the child, will then determine the fitness of the father, according a source familiar with Bahamian law.
If the DNA test shows that Birkhead is not the biological father, Stern will continue to have custody of Dannielynn for now because he is listed as her father on her birth certificate. If Birkhead is the father, he still may not necessarily get custody.
"Even though someone is the biological father, that doesn't mean he gets custody," the source tells PEOPLE.
Ron Rale, Stern's former law partner who also represented Smith in several of her legal battles, says that custody of the baby is a different issue from paternity. "[The court] might figure out who the biological father is, but that's not the end game for custody," he tells PEOPLE. "But the Bahamian court does have jurisdiction for deciding custody."
Meanwhile, Stern has hired a lawyer who said Monday he may sue any media outlets Stern thinks are implicating him in the overdose deaths of Smith and her son Daniel.
"The nightly television, tabloid and Internet trial of Mr. Stern in the court of public opinion based on sensational lies, speculation, rumor and gossip is over," Atlanta attorney Lin Wood said in a statement, according to the Associated Press.
"The medical examiner concluded that the death of Anna Nicole Smith was accidental. No one has produced any evidence to rebut this conclusion because no such evidence exists," said Wood, who also represented John and Patsy Ramsey after the death of their daughter JonBenet.
Reporting by HOWARD BREUER and SIOBHAN MORRISSEY