The bizarre story of the deaths of two infants - three years apart, but under nearly identical circumstances - is going before a jury at last as the murder trial of their mother, Paula Sims of Alton, begins in January 1990. A pale Sims watched Wednesday as lawyers finished three days of choosing jurors to hear the charges against her in the April 1989 death of her 6-week-old daughter, Heather Lee. The trial was moved to Peoria because of publicity in Madison County. Opening statements are set for today. Assistant State's Attorney Don W. Weber of Madison County and Sims' attorney, Donald E. Groshong of Alton, interviewed 88 people before finding four women and eight men to serve as jurors and three more men to serve as alternates. Even though Peoria is 150 miles north of Madison County, only two of the 88 interviewed said they had neither read nor heard about the case. One of those two was disqualified; the other is an alternate. Sims is not charged in the June 1986 death of her first daughter, 13-day-old Loralei Marie. But the jurors at this trial will learn of the haunting similarities between the two girls' deaths. Circuit Judge Andy Matoesian of Madison County agreed with Weber that Loralei's death could be considered limited evidence in Heather's killing. Sims could face the death penalty if convicted. Her husband, Robert Sims, 37, has not been charged. Authorities have called Robert Sims a suspect in both deaths; he is expected to testify. Paula Sims says her daughter was kidnapped by a masked gunman. An autopsy disclosed that the baby had been smothered and her body frozen before it was put in a trash bag and dumped in St. Charles County, Mo. The story is similar to the account she gave in the death of Loralei three years before. Groshong, her lawyer, asked prospective jurors this week if they knew about Sudden Infant Death Syndrome. That gave rise to speculation that he would argue that illness as the cause of death. To that issue, Weber said: ''Any port in a storm,'' suggesting that Groshong had little else to argue to the jury. Weber told potential jurors that his case would be entirely circumstantial, because he had no witnesses to the death. But he dismissed Paula Sims' accounts of both deaths as ''preposterous.'' Weber has said that he believed the motive in the slayings was the Simses' dislike of girls. He said evidence would show that they had made such comments to other people, and he noted that the Simses had a son - Randall, 2 - who was unharmed. Groshong has argued that nobody has evidence that Paula Sims killed anyone. In fact, Groshong told the jurors, nobody has proof that Heather died in Illinois. Such proof is needed to convict Sims in an Illinois court. In ruling Wednesday on motions, Matoesian: Held that Weber could not tell the jury of a blood test showing that Paula Sims had marijuana in her blood the night that Heather disappeared. Barred Weber from telling the jury that Sims' parents - Orville and Nylene Blew of Cottage Hills - had got a grant of immunity after they refused to testify before a grand jury. The grant forced the Blews to testify. Said he would rule today on Groshong's motion for the jury to be told about footprints near the trash barrel where Heather's body was found. Weber dismissed the footprints as unrelated to the case.