The 0$^+_2$ state in $^{34}$Si has been populated at the {sc Ganil/Lise3} facility through the $beta$-decay of a newly discovered 1$^+$ isomer in $^{34}$Al of 26(1) ms half-life. The simultaneous detection of $e^+e^-$ pairs allowed the determination of the excitation energy E(0$^+_2$)=2719(3) keV and the half-life T$_{1/2}$=19.4(7) ns, from which an electric monopole strength of $rho^2$(E0)=13.0(0.9)$times10^{-3}$ was deduced. The 2$^+_1$ state is observed to decay both to the 0$^+_1$ ground state and to the newly observed 0$^+_2$ state (via a 607(2) keV transition) with a ratio R(2$^+_1$$rightarrow0^+_1/2^+_1$$rightarrow0^+_2$)=1380(717). Gathering all information, a weak mixing with the 0$^+_1$ and a large deformation parameter of $beta$=0.29(4) are found for the 0$^+_2$ state, in good agreement with shell model calculations using a new {sc sdpf-u-mix} interaction allowing textit{np-nh} excitations across the N=20 shell gap.
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