We prove a Korn-type inequality in H̊(Curl;Ω,R{double-struck}^3×3) for tensor fields P mapping Ω to R{double-struck}^3×3. More precisely, let Ω⊂R{double-struck}³ be a bounded domain with connected Lipschitz boundary ∂ Ω. Then, there exists a constant c> 0 such that (1) c||P||_{L2(Ω,R{double-struck}3×3)}≤||symP||_{L2(Ω,R{double-struck}3×3)}+||CurlP||_{L2(Ω,R{double-struck}3×3}) holds for all tensor fields P∈H̊(Curl;Ω,R{double-struck}^3×3), i.e., all P∈H(Curl;Ω,R{double-struck}^3×3) with vanishing tangential trace on ∂ Ω. Here, rotation and tangential traces are defined row-wise. For compatible P, i.e., P=∇;v and thus Curl. P=0, where v∈H1(Ω,R{double-struck}³) are vector fields having components v_n, for which ∇;v_n are normal at ∂ Ω, the presented estimate (1) reduces to a non-standard variant of Korn's first inequality, i.e., c||∇;v||_{L2(Ω,R{double-struck}3×3)}≤||sym∇;v||_{L2(Ω,R{double-struck}3×3)}. On the other hand, for skew-symmetric P, i.e., sym. P=0, (1) reduces to a non-standard version of Poincaré's estimate. Therefore, since (1) admits the classical boundary conditions our result is a common generalization of these two classical estimates, namely Poincaré's resp. Korn's first inequality.