We propose and test a new whole-Teflon gate valve for handling droplets. The valve allows droplet plugs to pass through without disturbing them. This is possible due to the geometric design, the choice of material and lack of any pulses of flow generated by closing or opening the valve. The duct through the valve resembles a simple segment of tubing, without constrictions, change in lumen or side pockets. There are no extra sealing materials with different wettability or chemical resistance. The only material exposed to liquids is FEP Teflon, which is resistant to aggressive chemicals and fully biocompatible. The valve can be integrated into microfluidic systems: we demonstrate a complex system for culturing bacteria in hundreds of microliter droplet chemostats. The valve effectively isolates modules of the system to increase precision of operations on droplets. We verified that the valve allowed millions of droplet plugs to safely pass through, without any cross-contamination with bacteria between the droplets. The valve can be used in automating complex microfluidic systems for experiments in biochemistry, biology and organic chemistry.