To date, the only limit on graviton mass using galaxy clusters was obtained
by Goldhaber and Nieto in 1974, using the fact that the orbits of galaxy
clusters are bound and closed, and extend up to 580 kpc. From positing that
only a Newtonian potential gives rise to such stable bound orbits, a limit on
the graviton mass $m_g<10^{-29}$ eV was obtained (PRD 9,1119, 1974). Recently,
it has been shown that one can get closed bound orbits for a whole class of
non-Newtonian potentials (arXiv:1707.04937 and arXiv:1705.02444), thus
invalidating the main \emph{ansatz} used in Goldhaber and Nieto to obtain the
graviton mass bound. In order to obtain a revised estimate using galaxy
clusters, we use dynamical mass models of the Abell 1689 (A1689) galaxy cluster
to check their compatibility with a Yukawa gravitational potential. We assume
mass models for the gas, dark matter, and galaxies for A1689 from
arXiv:1703.10219 and arXiv:1610.01543, who used this cluster to test various
alternate gravity theories, which dispense with the need for dark matter. We
quantify the deviations in the acceleration profile using these mass models,
assuming a Yukawa potential and that obtained assuming a Newtonian potential,
by calculating the $\chi^2$ residuals between the two profiles. The 90\% c.l.
upper limit on the graviton mass corresponds to the minimum mass for which
$\Delta \chi^2>2.71$. Our estimated 90\% c.l. bound on the graviton mass
($m_g$) is thereby given by, $m_g < 1.64 \times 10^{-29}$ eV or in terms of the
graviton Compton wavelength, $\lambda_g>7.6 \times 10^{19}$ km.