An isosceles triangle has two sides with equal length and a base.
As a consequence, the two base angles are the same.
Now, there are two cases:
CASE 1: The 60° angle is the angle between the sides with equal length
In this case, we know that the two base angles must be the same, and together they form 180°-60°=120°. So, each of the base angle is a 60° angle, and so all three angles are 60° angles, and the triangle is equilateral
CASE 2: The 60° angle is one of the base angles
The other base angle must be a 60° angle as well. So, the two base angles form 120°, leaving 60° for the angle between the sides with equal length.
So, again, all three angles are 60°, and the triangle is equilateral.