An apparatus for accurately infusing fluids into a patient at specific rates over an extended period of time and the method for making same. The apparatus includes one or more dispensers, of a low profile, laminate or layered construction each having a stored energy source in the form of a distendable membrane or an elastomeric cellular mass, which in cooperation with the base, defined a fluid chamber having a fluid inlet and a fluid outlet. The apparatus further includes, in lieu of a rigid ullage, a high novel, conformable ullage made of yieldable materials. The conformable ullage uniquely conforms to the shape of elastomeric membrane as the membrane returns to its less distended configuration. This arrangement will satisfy even the most stringent delivery tolerance requirements and will elegantly overcome the limitations of materials selection encountered in devices embodying the rigid ullage construction. Additionally, with the novel ullage construction, the ullage can be located either between the base and the fluid to be delivered, or alternatively, can be located between the elastomeric membrane and the fluid to be delivered. Further, a plurality of subreservoirs can be associated with a single ullage thereby making it possible to incorporate a wide variety of delivery profiles within a single device.