Measurements have been made of the mean lifetimes of $\ensuremath{\mu}$-mesons in several heavy elements. Time lags between the arrival of a cosmic-ray meson in the target and its subsequent absorption---as signaled by the neutrons and gamma-rays following capture---are measured using large liquid scintillation counters and a chronotron timing circuit. The timing uncertainty is about 2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}9}$ sec, and the counting rates are such that a mean life can be determined to an accuracy of 10 percent in about a week's run at sea level. A short-lived component of the decay curves was found and identified as due to neutrons from proton-induced stars. Errors from this effect were avoided. Our latest results for the mean lives, in m\ensuremath{\mu}sec, are $\mathrm{Fe}163\ifmmode\pm\else\textpm\fi{}27, \mathrm{Hg}58\ifmmode\pm\else\textpm\fi{}4, \mathrm{Cu}116\ifmmode\pm\else\textpm\fi{}9, \mathrm{Pb}76\ifmmode\pm\else\textpm\fi{}4, \mathrm{Sb}99\ifmmode\pm\else\textpm\fi{}11, \mathrm{Bi}68\ifmmode\pm\else\textpm\fi{}5.$ The results are in agreement with the Wheeler ${{Z}_{\mathrm{eff}}}^{4}$ law up through Cu but disagree by a factor 3 for the heavier elements. They are in reasonable agreement with Kennedy's calculations (see following paper). The difference between the mean lives for Hg and Pb is also qualitatively predicted by Kennedy on the basis of a shell model of the nucleus. Our results, together with Kennedy's calculation for Pb, allow us to conclude that the $\ensuremath{\mu}$-meson-nucleon coupling constant has the same value, within about 25 percent, as recent values of the coupling constants of beta-decay and of the spontaneous disintegration of the $\ensuremath{\mu}$-meson.