In ride-sharing, commuters with similar itineraries share a vehicle for their trip. Despite its clear benefits in terms of reduced congestion, ride-sharing is not yet widely accepted. We propose a specific ride-sharing variant, where drivers are completely inflexible. This variant can form a competitive alternative against private transportation, due to the limited efforts that need to be made by drivers. However, due to this inflexibility, matching of drivers and riders can be substantially more complicated, compared to the situation where drivers can deviate. In this work, we propose a four-step procedure to identify the effect of such a ride-sharing scheme. We use a dynamic mesoscopic traffic simulator which computes departure-time choices and route choices for each commuter. The optimal matching of potential drivers and riders is obtained outside the simulation framework through an exact formulation of the problem. We evaluate the potential of this ride-sharing scheme on a real network of the Paris metropolitan area for the morning commute. We show that even with inflexible drivers and when only a small share of the population is willing to participate in the ride-sharing scheme, ride-sharing can alleviate congestion. Further improvements can be obtained by increasing the capacity of the vehicles or by providing small monetary incentives, but without jeopardizing the inflexibility of the drivers. Thereby, we show that ride-sharing can lead to fuel savings, CO\(_{2}\) emission reductions and travel time savings on a network level, even with a low participation rate.