Conventional gunshot detection systems can quickly and reliably detect gunshots in the area where the acoustic sensors are placed. This paper presents the detection of weak hunting gunshots using the short-term entropy of signal energy computed from acoustic signals in an open natural environment. Our research in this field was primarily aimed at detecting gunshots fired at close range with the usual acoustic intensity to protect wild elephants from poachers. The detection of weak gunshots can extend existing detection systems to detect more distant gunshots. The developed algorithm was optimized for the detection of gunshots in two categories of the surrounding sounds, short impulsive events and continuous noise, and tested in acoustic scenes where the power ratios between the weak gunshots and louder surroundings range from 0 dB to -14 dB. The overall accuracy was evaluated in terms of recall and precision. Depending on impulsive or noise sounds, binary detection was successful down to -8 dB or -6 dB; then, the efficiency decreases, but some very weak gunshots can still be detected at -13 dB. Experiments show that the proposed method has the potential to improve the efficiency and reliability of gunshot detection systems.