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Let ${𝒟}_{\lambda ,\mu }$ be the space of linear differential operators on weighted densities from ${ℱ}_{\lambda }$ to ${ℱ}_{\mu }$ as module over the orthosymplectic Lie superalgebra $\mathrm{𝔬𝔰𝔭}\left(3\right|2)$, where ${ℱ}_{\lambda }$, $ł\in ℂ$ is the space of tensor densities of degree $\lambda$ on the supercircle $S^{1|3}$. We prove the existence and uniqueness of projectively equivariant quantization map from the space of symbols to the space of differential operators. An explicite expression of this map is also given.