Behavior of the planar, burner-stabilized premixed flame that suffers volumetric heat loss, with its rate described by a linear function of temperature, is analyzed by activation energy asymptotics. Both its steady burning characteristics and its diffusional-thermal flame-front instability are investigated. The effect of volumetric heat loss is found to slow down the reaction so that the flame moves away from the burner. Consequently, the heat transfer from the flame to the burner is reduced and the flame temperature is increased. By continuously increasing the intensity of volumetric heat loss, the flame eventually is detached from the burner and becomes freely propagating. The stability analysis then reveals that the volumetric heat loss renders the flame to be less stable. similar to what is known for the freely propagating flame. The increase in the flame temperature has only a weak effect to stabilize the flame. It is primarily to compensate the decrease in the burning rate that is induced by the volumetric heat loss. In addition, the volumetric heat loss has a stronger impact on the reduction of stability for flames with a lower burner supply rate, that also is the burning rate. A critical burner supply rate exists, below which the flame turns absolutely unstable before detaching from the burner so that the freely propagating limit cannot be reached by increasing the intensity of volumetric heat loss. This study further supports the existence of the dual flame behavior experimentally observed by Spalding and Yumlu in that there exist two flame speeds for a heat transfer rate to the burner because stable flame exists, even for relatively small burner supply rates, when the volumetric heat loss is weak.