A dispensing closure for a container that holds a fluid, which closure can be tightly fitted into a container neck to ensure liquid-tightness of the container, comprising: a plug body arranged to be inserted into the container neck and having a central fluid passage and having a peripheral air passage, a double-valve diaphragm consisting of a flexible elastic material and arranged in a recess provided in an upper part of the plug body to cover the fluid passage and the air passage, and a plug cap for sealing the double-valve diaphragm within the plug body having one or more venting slots arranged circularly and axially aligned with the air passage of the plug body and having a central orifice axially aligned with the fluid passage of the plug body, wherein the double-valve diaphragm comprises two concentric valve sections with: a first valve section, and a second valve section, where the first valve section is a normally closed valve having a slit-type central orifice and which seals the central fluid passage of the plug body when in its resting position, and where the second valve section is an umbrella-type valve which is a normally closed valve and which seals the one or more venting slots of the plug cap when in its resting position, characterised in that the plug cap further has a ring seal provided between said venting slots and said central opening to prevent leakage of liquid when a tube or cannula is inserted into the valve located below the plug cap. a plug body arranged to be inserted into the container neck and having a central fluid passage and having a peripheral air passage, a double-valve diaphragm consisting of a flexible elastic material and arranged in a recess provided in an upper part of the plug body to cover the fluid passage and the air passage, and a plug cap for sealing the double-valve diaphragm within the plug body having one or more venting slots arranged circularly and axially aligned with the air passage of the plug body and having a central orifice axially aligned with the fluid passage of the plug body, wherein the double-valve diaphragm comprises two concentric valve sections with: a first valve section, and a second valve section, where the first valve section is a normally closed valve having a slit-type central orifice and which seals the central fluid passage of the plug body when in its resting position, and where the second valve section is an umbrella-type valve which is a normally closed valve and which seals the one or more venting slots of the plug cap when in its resting position, a first valve section, and a second valve section, where the first valve section is a normally closed valve having a slit-type central orifice and which seals the central fluid passage of the plug body when in its resting position, and where the second valve section is an umbrella-type valve which is a normally closed valve and which seals the one or more venting slots of the plug cap when in its resting position, characterised in that the plug cap further has a ring seal provided between said venting slots and said central opening to prevent leakage of liquid when a tube or cannula is inserted into the valve located below the plug cap.