American Promise Act of 2019 This bill provides certain eligible aliens with permanent residence status. The Department of Homeland Security (DHS) and Department of Justice shall cancel removal proceedings against an eligible alien who applies for an adjustment to permanent residence status. The alien must (1) have been continually physically present in the United States for at least three years; (2) not be inadmissible due to various reasons, such as those related to health or criminal activity; (3) have been eligible for temporary protected status (TPS) or deferred enforced departure (DED) status on a certain date; and (4) have not engaged in conduct that would disqualify the alien from such status. TPS and DED status allow nationals of certain countries to temporarily remain and work in the United States. DHS may not use information from applications to adjust status for immigration enforcement purposes. Such information may be shared with law enforcement only for limited purposes, such as to prevent fraudulent claims or to investigate felonies not related to immigration status. DHS shall establish a grant program for nonprofit organizations that assist individuals with certain immigration-related issues, including applications for adjustment of status under this bill. The bill imposes various requirements related to the application, such as fees, documents to be submitted, and biometric data for background checks. DHS may waive certain requirements for humanitarian concerns and other reasons.