I get roughly half Bevan's figure.
I make it 2.6 degrees at 300nm.
The solution is a bit long-winded though - so bound to be a mistake somewhere!
If you draw a tangent to the Earth's surface, and make the line 300nm long.
Join both ends of the tangent to the centre of the Earth.
You've got a right angle triangle.
The angle subtended at the centre of the Earth is atan(300/3443) = 5 degrees ( where 3443 is Earth radius approx.)
If you then calculate the length of the longer line from the Earth centre to the end of the tangent from 300/sin(5 degrees) =3456 nm.
The difference from the centre of the Earth to either end of the tangent being 3456 - 3443 = 13 nm.
And 13nm in 300 = 2.6 degrees.
The Earth's surface drops away from the tangent at roughly 0.5nm in 60 = 0.5 degrees.