Summary: Consider the system \[ u_t=\Delta u-\chi\nabla(u\nabla\nu),\qquad 0=\Delta\nu+(u-1)\quad \text{for }x\in\Omega,\;t>0, \tag{S} \] together with no-flux boundary conditions \({{\partial u}\over{\partial n}}={{\partial\nu}\over{\partial n}}=0\) for \(x\in\partial\Omega\), \(t>0\), and initial values \(u(x,0)= u_0(x)\), where \(u_0(x)\) is, say, a continuous and nonnegative function. Here, \(\Omega\) denotes a smooth and bounded open set in \(\mathbb{R}^2\), and \(\chi>0\). System (S) is a model to describe chemotaxis for a species (whose concentration is represented by \(u(x,t)\) under the action of a chemical (whose concentration is denoted by \(\nu(x,t)\)) which is secreted by the species organisms. We prove that if \(\Omega\) is a ball, there exist radial solutions of that problem, which concentrate into a Dirac mass in finite time. This fact is usually referred to as chemotactic collapse. The manner in which chemotactic collapse develops in our system is analyzed in detail.