Rope trick

d = 12 472km

r = 6 236km

p = d * pi =  $${\mathtt{12\,472}}{\mathtt{\,\times\,}}{\mathtt{\pi}} = {\mathtt{39\,181.943\: \!575\: \!571\: \!901\: \!270\: \!1}}$$

Let's add a meter:

(1m = 1/1000km)

$${\mathtt{39\,181.943\: \!58}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{1\,000}}}} = {\mathtt{39\,181.944\: \!58}}$$

And back to the diameter:

$${\frac{{\mathtt{39\,181.944\: \!58}}}{{\mathtt{\pi}}}} = {\mathtt{12\,472.000\: \!319\: \!719\: \!393\: \!786\: \!5}}$$

We have ~3.197 Meter more.

We can lift it now by 1.5985 Meter on each side. 

If h is the height it can be lifted, then we have:

Original circumference = pi*d  where d is diameter of the earth.

New circumference = pi*(d+2h)  since the new diameter is d+2h.

New - Old circumference:  pi*(d + 2h) - pi*d = 1m so that pi*2h = 1m  or h = 1/(2pi) m or h ≈ 0.159m

(xerxes, your next to last line should say "We have ∼0.3197 meters more).

.

sorry xerxes,answer is simply     1metre/2pi.

Here's the solution; let Earth's circumference be 2pi R. New circumference then is 2pi (R + delta R) where delta R is the height we want to find.

so 2pi R + 1 metre= 2pi(R+ deltaR),the 2pi R 's cancel out,giving 2pi delta R = 1 metre,so delta R,the height,is 1 metre /2pi. We did not need to know the Earth's circumference in order to solve the problem.