Families Under One Roof Act of 2019 This bill requires the Department of Housing and Urban Development (HUD) to make low-income housing assistance available for otherwise qualifying families with at least one member who has eligible immigration status. Specifically, a family with at least one member who has been affirmatively established as eligible for assistance, and no member who has been affirmatively established as ineligible due to immigration status, is eligible for prorated assistance based on the number of family members with established eligibility. Current law specifies that any assistance to such a mixed-status family must be prorated, and HUD has generally allowed such a family to receive assistance in this manner. However, a proposed HUD rule from May 10, 2019, would generally prohibit assistance to mixed-status families. The bill further specifies that, if such assistance for an individual is to be terminated on the basis of immigration status, it may instead be continued if necessary to avoid the division of a family in which any member has eligible immigration status. Under current law, such assistance may be continued under these circumstances only if the head of household or the individual's spouse has eligible immigration status.