Answer
The breaking will occur at the lowest point of the circle.
Work Step by Step
We can write an expression for the force on the stone from the cord at the top of the circle:
$F+mg = \frac{mv^2}{r}$
$F = \frac{mv^2}{r} - mg$
We can write an expression for the force on the stone from the cord at the bottom of the circle:
$F-mg = \frac{mv^2}{r}$
$F = \frac{mv^2}{r} + mg$
Since the required force is greater at the bottom of the circle, the breaking will occur at the lowest point of the circle.
