  The ground states of the frustrated pyrochlore oxide
Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$, sensitively depending on the small
off-stoichiometry parameter $x$, have been studied by specific heat
measurements using well characterized samples. Single crystal
Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ boules grown by the standard floating zone
technique are shown to exhibit concentration ($x$) gradient. This
off-stoichiometry parameter is determined by precisely measuring the lattice
constant of small samples cut from a crystal boule. Specific heat shows that
the phase boundary of the electric quadrupolar state has a dome structure in
the $x$-$T$ phase diagram with the highest $T_{\text{c}} \simeq 0.5$ K at about
$x = 0.01$. This phase diagram suggests that the putative U(1) quantum
spin-liquid state of Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ exists in the range
$x<x_{\text{c}} \simeq -0.0025 $, which is separated from the quadrupolar state
via a first-order phase-transition line $x=x_{\text{c}}$.
