We transfer an optical frequency over $251\mathrm{km}$ of optical fiber with a residual instability of $6\times {10}^{-19}$ at $100\mathrm{s}$ . This instability and the associated timing jitter are limited fundamentally by the noise on the optical fiber and the link length. We give a simple expression for calculating the achievable instability and jitter over a fiber link. Transfer of optical stability over this long distance requires a highly coherent optical source, provided here by a cw fiber laser locked to a high finesse optical cavity. A sufficient optical carrier signal is delivered to the remote fiber end by incorporating two-way, in-line erbium-doped fiber amplifiers to balance the $62\mathrm{dB}$ link loss.