Prevailing methods for measuring sensitive outcomes confront researchers with an inherent bias-variance trade-off: direct questioning is prone to a sensitivity bias, while indirect methods such as list experiments are substantially less precise. We introduce the ballot-bag, a novel technique that relaxes this trade-off by mitigating bias in direct questioning while improving precision over indirect methods. In a field experiment in Egypt, where direct questions on irregular migration are biased, ballot-bag estimates closely align with those from a list experiment but exhibit significantly lower variance. Consequently, treatment effects are highly significant via the ballot-bag and not via the list experiment.