  The recently discovered kagome metal series $A$V$_3$Sb$_5$ ($A$=K, Rb, Cs)
exhibits topologically nontrivial band structures, chiral charge order and
superconductivity, presenting a unique platform for realizing exotic electronic
states. The nature of the superconducting state and the corresponding pairing
symmetry are key questions that demand experimental clarification. Here, using
a technique based on the tunneling diode oscillator, the magnetic penetration
depth $\Delta\lambda(T)$ of CsV$_3$Sb$_5$ was measured down to 0.07 K. A clear
exponential behavior in $\Delta\lambda(T)$ with marked deviations from a $T$ or
$T^2$ temperature dependence is observed at low temperatures, indicating a
deficiency of nodal quasiparticles. Temperature dependence of the superfluid
density and electronic specific heat can be described by two-gap $s$-wave
superconductivity, consistent with the presence of multiple Fermi surfaces in
CsV$_3$Sb$_5$. These results evidence nodeless superconductivity in
CsV$_3$Sb$_5$ under ambient pressure, and constrain the allowed pairing
symmetry.
