  We report the discovery of superconductivity in binary alloy
Sn$_{1-x}$Sb$_{x}$ with $x$ in the range of 0.43 to 0.6, which possesses a
modulated rhombohedral structure due to the incommensurate ordering of Sn and
Sb layers along the $c$-axis. The specific heat measurements indicate a weakly
coupled, fully gapped superconducting state in this homogeneity range with a
maximum bulk $T_{\rm c}$ of 1.58 K at $x$ = 0.46, though the electronic
specific heat and Hall coefficients remain nearly $x$-independent. The
nonmonotonic dependence of the bulk $T_{\rm c}$ is discussed in relation to the
effects of Sb-layer intercalation between the [Sn$_{4}$Sb$_{3}$] seven-layer
lamellae that are the essential building block for superconductivity. On the
other hand, a zero-resistivity transition is found to take place well above the
bulk superconducting transition, and the corresponding $T_{\rm c}$ increases
monotonically with $x$ from 2.06 K to 3.29 K. This contrast, together with the
uniform elements distribution revealed by energy dispersive x-ray mapping,
implies that the resistive transition is due to the strain effect at the grain
boundary rather than the compositional inhomogeneity. The first-principles
calculations on the representative composition Sn$_{4}$Sb$_{3}$ ($x$ = 0.43)
indicate that it is topologically nontrivial similar to Sb, but with different
Z$_{2}$ invariants (0;111). Our results not only identify a new superconducting
region in the Sn-Sb phase diagram, but also provide a viable platform to study
the interplay between structural modulation, nontrivial band topology and
superconductivity.
