Using a combination of laser–plasma interactions and magnetic confinement configurations, a conceptual fusion reactor is proposed in this paper. Our reactor consists of the following: (1) A background plasma of boron11 and hydrogen ions, plus electrons, is generated and kept for a certain time, with densities of the order of a mg/cm3 and temperatures of tens of eV. Both the radiation level and the plasma thermal pressure are thus very low. (2) A plasma channel is induced in a solid target by irradiation with a high power laser that creates a very intense shock wave. This mechanism conveys the acceleration of protons in the laser direction. The mechanisms must be tuned for the protons to reach a kinetic energy of 300–1200 keV where the pB11 fusion cross section is significantly large (note that this value is not a temperature). (3) Those ultra-fast protons enter the background plasma and collide with boron11 to produce three alphas. Fusion born alphas collide with protons of the plasma and accelerate them causing a chain reaction. (4) A combination of an induction current and a magnetic bottle keeps the chain reaction process going on, for a pulse long enough to get a high energy gain. (5) Materials for the background plasma and the laser target must be replaced for starting a new chain reaction cycle.