  In the six-dimensional Kaluza-Klein model with the multidimensional
cosmological constant $\Lambda_6$, we obtain the black brane with spherical
compactification of the internal space. The matter source for this exact
solution consists of two parts. First, it is a fine-tuned homogeneous perfect
fluid which provides spherical compactification of the internal space. Second,
it is a gravitating massive body with the dustlike equation of state in the
external space and tension $\hat p_1=-(1/2)\hat\varepsilon$ in the internal
space. This solution exists both in the presence and absence of $\Lambda_6$. In
the weak-field approximation, we also get solutions of the linearized Einstein
equations for the model with spherical compactification. Here, the gravitating
matter source has the dustlike equation of state in the external space and an
arbitrary equation of state $\hat p_1=\Omega \hat\varepsilon$ in the internal
space. In the case $\Lambda_6>0$ and $\Omega\neq -1/2$, these approximate
solutions tend asymptotically to the weak-field limit of the exact black brane
solution. Both the exact and asymptotic black branes satisfy the gravitational
experiments at the same level of accuracy as general relativity.
