In a letter, Cupich said the alleged abuse happened while Father James McIlhone was the associate pastor of Santa Maria del Popolo Parish in Mundelein in the 1980s.
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"Allegations are claims that have not been proven and for this reason, there needs to be a full investigation," said a letter to the parish.
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Cardinal Cupich said the priest has been asked to move away from the Saint Edward parish where he has been living during retirement. Letters were sent to both the St. Edward and Santa Maria del Popolo parishes Saturday.
The person who made the allegation has been offered victim support services, and the archdiocese has begun its investigation of these matters, according to the Archdiocese of Chicago.
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"The Archdiocese of Chicago takes all allegations of sexual misconduct seriously and encourages anyone who feels they have been sexually abused by a priest, deacon, religious or lay employee to come forward," the letter continued.
The allegations have also been reported to the Lake County State's Attorney.