The Scapular Promise and the Sabbatine Privilege are two separate things.  The Sabbatine Privilege is the doctrine that those who practice certain disciplines accompanying their devotion to Our Lady will be delivered by her from Purgatory on the First Saturday after their death.  As early as 1613, the Holy See determined that the decree establishing the Sabbatine Privilege was unfounded and admonished the Carmelite Order not to preach this doctrine.  

What does the Church permit us to believe today related to this doctrine?  To answer this question, we turn to a declaration from the Congregation of Indulgences on July 4, 1908:

“It is permitted to the Carmelite Fathers to preach that the Christian people may piously believe in the help which the souls of brothers and members, who have departed this life in charity, have worn throughout life the scapular, have ever observed chastity, have recited the Little Hours [of the Blessed Virgin], or, if they cannot read, have observed the fast days of the Church, and have abstained from flesh meat on Wednesdays and Saturdays (except when Christmas falls on such days), may derive after death—especially on Saturdays, the day consecrated by the Church to the Blessed Virgin—through the unceasing intercession of Mary, her pious petitions, her merits, and her special protection.”