Method and system are provided for the automated assembly of parts such as plastic parts into assemblies wherein assembly pallets loaded with their corresponding assemblies are stored and vertically indexed in an accumulator station to allow a previously applied material such as an adhesive to cure. Consequently, the system takes up little floor space. Entrance, dispensing and assembly stations, together with the accumulator station each has an enclosed chamber which is pressurized with an ambient at subatmospheric pressure in order to evacuate the enclosed chambers so that fumes from the adhesive utilized in securing the parts together are not allowed to escape at the stations during assembly of the parts.