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We consider the Keller-Segel system with gradient dependent chemotactic sensitivity $$\displaylines{ u_t =\Delta u-\nabla\cdot(u|\nabla v|^{p-2}\nabla v),\quad x\in\Omega,\; t>0,\cr v_t =\Delta v-v+u,\quad x\in\Omega,\; t>0,\cr \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial\nu}=0,\quad x\in\partial\Omega,\; t>0,\cr u(x,0)=u_0(x),\quad v(x,0)=v_0(x), \quad x\in\Omega }$$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$, $n\geq2$. We shown that for all reasonably regular initial data $u_0\geq 0$ and $v_0\geq0$, the corresponding Neumann initial-boundary value problem possesses a global weak solution which is uniformly bounded provided that $1<p<n/(n-1)$.