A

To find the period of a pendulum at the South Pole, we need to consider the effect of Earth's rotation, which influences the pendulum's effective gravitational acceleration at different latitudes.

Step 1: Understand the formula for the period of a simple pendulum, given by:
$$T = 2\pi \sqrt{\frac{L}{g}}$$
where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.

Step 2: At the equator, the effective gravity is influenced by Earth's rotation. The effective gravitational acceleration (g') can be expressed as:
$$g' = g - R\omega^2 \cos^2\theta$$
where R is the Earth's radius, \omega is the angular velocity of the Earth, and \theta is the latitude.
For the equator (\theta = 0), g' is simply g (approximately 9.81 m/s²).

Step 3: At the South Pole (\theta = 90\degree), the effect of Earth's rotation does not apply (since cos(90°) = 0). Therefore, at the South Pole, the effective gravitational acceleration remains:
$$g' = g$$
Thus, the period T at the South Pole will also be influenced by gravity only.

Step 4: Since the period at the equator was exactly 1.00 sec, the period at the South Pole will also be:
$$T_{polar} = 2\pi \sqrt{\frac{L}{g}}$$
Since g does not change between the equator and the South Pole, the period remains the same.

Conclusion: Therefore, the period of the pendulum at the South Pole remains 1.00 sec as well.
Therefore, the correct answer is A.