For a real smooth algebraic curve A ⊂ (C^*)², the amoeba A ⊂ R² is the image of A under the map Log : (x,y) → ( log |x|, log |y| ). We describe a universal bound for the total curvature of the real amoeba A_RA and we prove that this bound is reached if and only if the curve A is a simple Harnack curve in the sense of Mikhalkin.