With integrated trade and financial markets, a collapse in aggregate demand in a large country can cause "natural real interest rates" to fall below zero in all countries, giving rise to a global "liquidity trap." This paper explores the optimal policy response to this type of shock, when governments cooperate on both fiscal and monetary policy. Adjusting to a large negative demand shock requires raising world aggregate demand, as well as redirecting demand towards the source (home) country. ; The key feature of demand shocks in a liquidity trap is that relative prices respond perversely. A negative shock causes an appreciation of the home terms of trade, exacerbating the slump in the home country. At the zero bound, the home country cannot counter this shock. Because of this, it may be optimal for the foreign policymaker to raise interest rates. ; Strikingly, the foreign country may choose to have a positive policy interest rate, even though its natural real interest rate is below zero. A combination of relatively tight monetary policy in the foreign country combined with substantial fiscal expansion in the home country achieves the desired mix in terms of the level and composition of world expenditure. Thus, in response to conditions generating a global liquidity trap, there is a critical mutual interaction between monetary and fiscal policy.