This paper analyzes automated distributive negotiation where agents have firm deadlines that are private information. The agents are allowed to make and accept offers in any order in continuous time. We show that the only sequential equilibrium outcome is one where the agents wait until the first deadline, at which point that agent concedes everything to the other. This holds for pure and mixed strategies. So, interestingly, rational agents can never agree to a nontrivial split because offers signal enough weakness of bargaining power (early deadline) so that the recipient should never accept. Similarly, the offerer knows that it offered too much if the offer gets accepted: the offerer could have done better by out-waiting the opponent. In most cases, the deadline effect completely overrides time discounting and risk aversion: an agent's payoff does not change with its discount factor or risk attitude. Several implications for the design of negotiating agents are discussed. We also present an effective protocol that implements the equilibrium outcome in dominant strategies.