Using the diameter of the Earth at the equator of 7926.28 miles:

260 miles across the surface of the earth equates to:

(260 / (7926.28 * pi)) * 360 = 3.758863 degrees (260 miles divided by the circumference of earth in degrees - probably should have done it in radians but this still works)

Assuming the mountain is perpendicular to the earth, the distance from the centre of the earth to the tip of the mountain will be:

(7926.28 / 2) / cos (3.758863) = 3971.6839 miles (working out the length of the hypotenuse of a right angle triangle)

So the mountatain height will be

3971.6839 - (7926.28 / 2) = 8.5439 miles

Can draw a diagram if necessary...

Edit: Bah... Used the diameter in the preview on a google search which appears to be wrong. You can recalculate if you want.
http://www.google.com.au/search?sourceid...=earth+diameter

Looks like it should be 7926.41 miles here.
http://geography.about.com/library/faq/blqzdiameter.htm

Wikipedia has it in km but equates to 7926.55 miles diameter