The Equator’s Extra Metre

Imagine Earth as a perfect sphere. A rope is pulled tightly around the equator, touching the ground at every point. The rope is then cut, extended by exactly one metre, and joined again. Instead of leaving the extra length bunched in one place, the rope is adjusted so that it forms a larger circle concentric with Earth and remains the same height above the surface everywhere. How large is the uniform gap between the rope and the ground? Assume the rope does not stretch, and ignore mountains, oceans, and Earth’s slight flattening.