How strongly can the reduction in bureaucracy and red tape at the border increase trade? To address this, we study an ambitious trade facilitation policy - the complete abolition of internal customs controls - in the Eurasian Customs Union. Using a structural gravity model with high-dimensional fixed effects, we find evidence that non-tariff trade facilitation strongly and robustly increases the number of newly traded products. The value of trade, however, reacts more strongly to a traditional tariff-driven diversion of trade towards union members. Trade facilitation policies can diversify bilateral trade, highlighting the importance of trade analysis at the extensive margin.