Rescue teams went to the aid of the distressed whale after discovering it was in difficulty at Torry Bay, near Crombie Point in Fife, yesterday morning.
They hoped to refloat the animal but vets decided the four-metre-long whale should be put down after finding it was not in a condition to go back into the water.
Tissue samples are being taken to help work out what caused the whale to become beached. Its body will be removed from the area.
The death comes three days after a pilot whale died when it became stranded in shallow waters in the Firth of Forth. At this stage, experts do not believe the deaths are linked.
Before the minke whale was put down, a coastguard team, fire crews and members of the British Divers Marine Life Rescue (BDMLR) were at the scene assessing the situation and doing what they could to ease the minke's distress.
A spokeswoman for the BDMLR said the animal was showing signs of distress and was moving quite a lot in the mud.
"We had hoped for a rescue but it soon became clear that we weren't going to be able to, so it was just a case of keeping it comfortable," she said.
"It was clear it wasn't in a condition to be put back in the water so the vets made the decision to have it put to sleep.
"Samples are being taken at the moment. "
The spokeswoman said the two cases did not appear to be linked. She said: "They'll have come from different areas."