This is the problem that led to the classical problem of elliptic integrals. However, you can solve your problem numerically using, for example, the 'integral' function. If your ellipse has its major and minor axes aligned with the coordinate axes and if you assume that its center is as the coordinate origin, the ellipse has the equation
x^2/a^2 + y^2/b^2 = 1
where a and b are the major and minor semi-axis lengths. Suppose the x coordinate at point A is x1 and at B it is x2. Then the arc length between A and B would be
integral of sqrt(1+(dy/dx)^2) with respect to x from x = x1 to x = x2
It can be shown that this integrand is given by the expression:
sqrt((a^4-(a^2-b^2)*x^2)/(a^4-a^2*x^2))
I have assumed here that points A and B are both in the upper part of the ellipse. For other parts of the ellipse the necessary adjustment is easy to make.
