A process for the hydrotreatment of a fuel gas stream containing up to 15% olefins comprises the steps of introducing the fuel gas stream into at least one co-current reactor, where the stream is split into two flow fractions, of which one fraction is routed through an olefin treatment section, while the other fraction is routed through another section, subjecting the sections to heat exchange, combining the two flows, thereby equalizing temperatures and compositions, cooling the combined flow over a heat exchanger and reacting the combined flow to equilibrium in an adiabatic hydrotreatment reactor. A second co-current reactor with intercooling arranged in series after the first cocurrent reactor and before the final adiabatic reactor is used if the fuel gas stream contains more than 8% olefins.