Summary: ``The aim of this note is to study Čebyšëv $\rm JB^*$-subtriples of general $\rm JB^*$-triples. It is established that if $F$ is a non-zero Čebyšëv $\rm JB^*$-subtriple of a $\rm JB^*$-triple $E$, then exactly one of the following statements holds: \roster \item"(a)" $F$ is a rank one $\rm JBW^*$-triple with $\dim(F)\geq 2$ (i.e. a complex Hilbert space regarded as a type 1 Cartan factor). Moreover, $F$ may be a closed subspace of arbitrary dimension and $E$ may have arbitrary rank; \item"(b)" $F=\Bbb Ce$, where $e$ is a complete tripotent in $E$; \item"(c)" $E$ and $F$ are rank two $\rm JBW^*$-triples, but $F$ may have arbitrary dimension; \item"(d)" $F$ has rank greater or equal than three and $E = F$.'' \endroster