  The long-lived $^{60}$Fe (with a half-life of 2.62 Myr) is a crucial
diagnostic of active nucleosynthesis in the Milky Way galaxy and in supernovae
near the solar system. The neutron-capture reaction
$^{59}$Fe(n,$\gamma$)$^{60}$Fe on $^{59}$Fe (half-life = 44.5 days) is the key
reaction for the production of $^{60}$Fe in massive stars. This reaction cross
section has been previously constrained by the Coulomb dissociation experiment,
which offered partial constraint on the $E$1 $\gamma$-ray strength function but
a negligible constraint on the $M$1 and $E$2 components. In this work, for the
first time, we use the surrogate ratio method to experimentally determine the
$^{59}$Fe(n,$\gamma$)$^{60}$Fe cross sections in which all the components are
included. We derived a Maxwellian-averaged cross section of 27.5 $\pm$ 3.5 mb
at $kT$= 30 keV and 13.4 $\pm$ 1.7 mb at $kT$= 90 keV, roughly 10 - 20% higher
than previous estimates. We analyzed the impact of our new reaction rates in
nucleosynthesis models of massive stars and found that uncertainties in the
production of $^{60}$Fe from the $^{59}$Fe(n,$\gamma$)$^{60}$Fe rate are at
most of 25%. We conclude that stellar physics uncertainties now play a major
role in the accurate evaluation of the stellar production of $^{60}$Fe.
