  The recently discovered kagome superconductors $A$V$_3$Sb$_5$ ($A$ = K, Rb,
Cs) possess a unique band structure with van Hove singularities and Dirac
dispersions, in which unusual charge-density-wave (CDW) orders with
time-reversal and rotational symmetry breaking have been reported. One of the
most crucial unresolved issues is identifying the symmetry of the
superconductivity that develops inside the CDW phase. Theory predicts a variety
of unconventional superconducting symmetries, including exotic states with
chiral and topological properties accompanied by a sign-changing
superconducting gap. Experimentally, however, the phase information on the
superconducting gap in $A$V$_3$Sb$_5$ is still lacking. Here we report the
electron irradiation effects in CsV$_3$Sb$_5$ using introduced impurities as a
phase-sensitive probe of superconductivity. Our magnetic penetration depth
measurements reveal that with increasing impurities, a highly anisotropic
fully-gapped state changes gradually to an isotropic full-gap state without
passing through a nodal state. Furthermore, transport measurements under high
pressure show that the double superconducting dome in the pressure-temperature
phase diagram survives against sufficient impurities. These results are strong
bulk evidence that CsV$_3$Sb$_5$ is a non-chiral, anisotropic $s$-wave
superconductor with no sign change both at ambient and high pressure, which
provides a clue to understanding the relationship between CDW and
superconductivity in kagome superconductors.
