  We report on measurements of the sound velocity and attenuation in a single
crystal of the candidate quantum- spin-ice material Yb$_2$Ti$_2$O$_7$ as a
function of temperature and magnetic field. The acoustic modes couple to the
spins magneto-elastically and, hence, carry information about the spin
correlations that sheds light on the intricate magnetic phase diagram of
Yb$_2$Ti$_2$O$_7$ and the nature of spin dynamics in the material.
Particularly, we find a pronounced thermal hysteresis in the acoustic data with
a concomitant peak in the specific heat indicating a possible first-order phase
transition at about $0.17$ K. At low temperatures, the acoustic response to
magnetic field saturates hinting at the development of magnetic order.
Furthermore, mean-field calculations suggest that Yb$_2$Ti$_2$O$_7$ undergoes a
first-order phase transition from a cooperative paramagnetic phase to a
ferromagnet below $T\approx 0.17$ K.
