The elastic and nonlinear acoustic properties of an antiferromagnetic fcc iron-manganese alloy single crystal with composition 60 at. % Fe have been studied with the ultrasonic pulse-echo-overlap technique. Velocity measurements of the three ultrasonic modes that can be propagated along the [110] direction have been made between 4.2 and 750 K to obtain all three independent elastic-stiffness-tensor components ${\mathit{C}}_{\mathit{I}\mathit{J}}$ and the adiabatic bulk modulus ${\mathit{B}}^{\mathit{S}}$ as a function of temperature. At 293 K the elastic stiffnesses are ${\mathit{C}}_{11}$=170 GPa, ${\mathit{C}}_{12}$=98 GPa, and ${\mathit{C}}_{44}$=142 GPa; hence ${\mathit{C}}_{11}$ and ${\mathit{B}}^{\mathit{S}}$=123 GPa are small, conforming with recognized trends for 3d transition-metal alloys. The N\'eel temperature ${\mathit{T}}_{\mathit{N}}$, assessed from electrical-resistance measurements and the steplike decrease in the temperature dependence of the shear modulus (${\mathit{C}}_{11}$-${\mathit{C}}_{12}$)/2, is 467 K. The contributions of antiferromagnetic ordering to ${\mathit{C}}_{\mathit{L}}$[=(${\mathit{C}}_{11}$+${\mathit{C}}_{12}$+2${\mathit{C}}_{44}$)/2], (${\mathit{C}}_{11}$-${\mathit{C}}_{12}$)/2, and ${\mathit{B}}^{\mathit{S}}$ are negative for all temperatures, while ${\mathit{C}}_{44}$ is stiffened slightly. Measurements of the hydrostatic pressure dependences of the velocities of ultrasonic modes propagated along the [110] direction have been used to obtain the hydrostatic pressure derivatives (\ensuremath{\partial}${\mathit{C}}_{\mathit{I}\mathit{J}}$/\ensuremath{\partial}P${)}_{\mathit{P}=0}$ of the elastic-stiffness-tensor components as a function of temperature in the antiferromagnetic Invar state.At 293 K, (\ensuremath{\partial}${\mathit{C}}_{11}$/\ensuremath{\partial}P${)}_{\mathit{P}=0}$, (\ensuremath{\partial}${\mathit{C}}_{12}$/\ensuremath{\partial}P${)}_{\mathit{P}=0}$, (\ensuremath{\partial}${\mathit{C}}_{44}$/\ensuremath{\partial}P${)}_{\mathit{P}=0}$, and (\ensuremath{\partial}${\mathit{B}}^{\mathit{S}}$/\ensuremath{\partial}P${)}_{\mathit{P}=0}$ are 10.1\ifmmode\pm\else\textpm\fi{}0.2, 7.1\ifmmode\pm\else\textpm\fi{}0.2, 3.84\ifmmode\pm\else\textpm\fi{}0.06, and 8.1\ifmmode\pm\else\textpm\fi{}0.2, respectively. To establish the vibrational anharmonicity of the long-wavelength acoustic modes, the results obtained for the ${\mathit{C}}_{\mathit{I}\mathit{J}}$ and (\ensuremath{\partial}${\mathit{C}}_{\mathit{I}\mathit{J}}$/\ensuremath{\partial}P${)}_{\mathit{P}=0}$ have been used to calculate the corresponding Gr\"uneisen parameters. The observation that at 293 K the mean long-wavelength acoustic-mode Gr\"uneisen parameter \ensuremath{\gamma}${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{\mathrm{el}}$(=2.12) is much larger than the thermal Gr\"uneisen parameter ${\ensuremath{\gamma}}^{\mathrm{th}}$(=0.81) shows that the vibrational anharmonicities of the long-wavelength acoustic modes are, on average, substantially larger than those of phonons having larger wave vectors. Magnetoelastic contributions to the vibrational anharmonicity of the long-wavelength acoustic modes of antiferromag- netic ${\mathrm{Fe}}_{60}$${\mathrm{Mn}}_{40}$ and ferromagnetic ${\mathrm{Fe}}_{72}$${\mathrm{Pt}}_{28}$ and ${\mathrm{Fe}}_{65}$${\mathrm{Ni}}_{35}$ Invar alloys are discussed in relation to the recently developed theoretical explanation of the behavior of Invar alloys based on fixed-spin-moment calculations of the total energy for low-spin and high-spin configurations.