A

To find the minimum velocity at the bottom of the circle for a body attached to a string of length \( \ell \), we need to analyze the forces acting on the body at the highest point of the vertical circle.

Step 1: At the highest point, the centripetal force must be provided by the weight of the body. This can be expressed as:
\[ mg = \frac{mv^2}{r} \]
where \( m \) is the mass, \( g \) is the acceleration due to gravity, \( v \) is the velocity at the highest point, and \( r \) is the radius of the circle (which is equal to the length of the string, \( \ell \)).

Step 2: Rearranging the equation gives:
\[ v^2 = rg
\] Hence, at the highest point, the minimum velocity \( v_H \) required is:
\[ v_H = \sqrt{g\ell} \]

Step 3: Now we need to find the velocity at the bottom of the circle using energy conservation. The potential energy at the bottom is zero and at the top it is given by:
\[ PE_H = mg(2\ell) \]
Step 4: The kinetic energy at the bottom is:
\[ KE_B = \frac{1}{2}mv_B^2 \] and at the top is:
\[ KE_H = \frac{1}{2}mv_H^2 \]
By conservation of energy:
\[ KE_B + PE_B = KE_H + PE_H \]
Since the potential energy at the bottom (PE_B) is zero, we have:
\[ \frac{1}{2}mv_B^2 = \frac{1}{2}mv_H^2 + mg(2\ell) \]
Step 5: This leads to:
\[ v_B^2 = v_H^2 + 4g\ell \]
Substituting \( v_H = \sqrt{g\ell} \):
\[ v_B^2 = (g\ell) + 4g\ell = 5g\ell \]
Step 6: Therefore, the minimum velocity at the bottom is:
\[ v_B = \sqrt{5g\ell} \]
Thus, the answer is \( \sqrt{5g\ell} \). Therefore, the correct answer option is A.