In linear accelerator free-electron laser facilities, energy spread is an important parameter reflecting the quality of the beam. The traditional measurement method, which uses a bending magnet, fluorescent screens and cameras, is an interceptive measurement method. The beam hitting the screen cannot be utilized during the measurement, and the measured data is the average of multiple macro pulses. Moreover, when using the measurement results of this method for beam tuning, the commissioning may take a long time since the fluorescent screens need to be repeatedly inserted and raised from the beam trajectory, thus new non-interceptive measurement methods need to be developed. This paper implements such a method using a single 4-stripline beam position monitor in conjunction with the emittance measurement method. This method is able to measure the energy spread of each macro beam pulse without inserting extra equipment. The measurement results are in agreement with those of the conventional interceptive measurement method. It is indicated by the error analysis that in order to reduce the measurement systematic error, it is necessary to make the beam pass through the center of the beam position monitor.