Secondary school completion in sub-Saharan Africa is the lowest in the world. Given the multiple constraints households face, the scope of purely demand-side interventions to narrow the completion gap may be large. This paper quantifies this scope by studying how jointly relaxing key demand-side constraints affects learning, graduation, and the labor market entry of talented and economically disadvantaged students in Uganda. The study randomizes access to a “big push” scholarship program covering fees, school placement, school inputs, and a cash transfer equivalent to 50 percent of the adult wage. The program raises test scores and pushes completion rates to high-income country levels; it halves fertility and increases the share of women who work while studying. Students‘ families of origin also benefit by becoming wealthier and happier. Results show that relaxing demand-side constraints benefits students and doubles as an antipoverty intervention for their households.