  The 1-arm exponent $\rho$ for the ferromagnetic Ising model on $\mathbb{Z}^d$
is the critical exponent that describes how fast the critical 1-spin
expectation at the center of the ball of radius $r$ surrounded by plus spins
decays in powers of $r$. Suppose that the spin-spin coupling $J$ is
translation-invariant, $\mathbb{Z}^d$-symmetric and finite-range. Using the
random-current representation and assuming the anomalous dimension $\eta=0$, we
show that the optimal mean-field bound $\rho\le1$ holds for all dimensions
$d>4$. This significantly improves a bound previously obtained by a
hyperscaling inequality.
