A Berge-$K_4$ in a triple system is a configuration with four vertices $v_1,v_2,v_3,v_4$ and six distinct triples ${e_{ij}: 1le i< j le 4}$ such that ${v_i,v_j}subset e_{ij}$ for every $1le i<jle 4$. We denote by $cal{B}$ the set of Berge-$K_4$ configurations. A triple system is $cal{B}$-free if it does not contain any member of $cal{B}$. We prove that the maximum number of triples in a $cal{B}$-free triple system on $nge 6$ points is obtained by the balanced complete $3$-partite triple system: all triples ${abc: ain A, bin B, cin C}$ where $A,B,C$ is a partition of $n$ points with $$leftlfloor{nover 3}rightrfloor=|A|le |B|le |C|=leftlceil{nover 3}rightrceil.$$
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