In this paper we report the impact of uniaxial strain $varepsilon$ applied along the crystalline $a$ axis on the newly discovered kagome superconductor CsV$_3$Sb$_5$. At ambient conditions, CsV$_3$Sb$_5$ shows a charge-density wave (CDW) transition at $T_{rm CDW}=94.5$ K and superconducts below $T_C = 3.34$ K. In our study, when the uniaxial strain $varepsilon$ is varied from $-0.90%$ to $0.90%$, $T_C$ monotonically increases by $sim 33%$ from 3.0 K to 4.0 K, giving rise to the empirical relation $T_C (varepsilon)=3.4+0.56varepsilon+0.12varepsilon^2$. On the other hand, for $varepsilon$ changing from $-0.76%$ to $1.26%$, $T_{rm CDW}$ decreases monotonically by $sim 10%$ from 97.5 K to 87.5 K with $T_{rm CDW}(varepsilon)=94.5-4.72varepsilon-0.60varepsilon^2$. The opposite response of $T_C$ and $T_{rm CDW}$ to the uniaxial strain suggests strong competition between these two orders. Comparison with hydrostatic pressure measurements indicate that it is the change in the $c$-axis that is responsible for these behaviors of the CDW and superconducting transitions, and that the explicit breaking of the sixfold rotational symmetry by strain has a negligible effect. Combined with our first-principles calculations and phenomenological analysis, we conclude that the enhancement in $T_C$ with decreasing $c$ is caused primarily by the suppression of $T_{rm CDW}$, rather than strain-induced modifications in the bare superconducting parameters. We propose that the sensitivity of $T_{rm CDW}$ with respect to the changes in the $c$-axis arises from the impact of the latter on the trilinear coupling between the $M_1^+$ and $L_2^-$ phonon modes associated with the CDW. Overall, our work reveals that the $c$-axis lattice parameter, which can be controlled by both pressure and uniaxial strain, is a powerful tuning knob for the phase diagram of CsV$_3$Sb$_5$.
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