Permanent connectivity to the Internet has become the defacto standard in the second decade of the 21st century. However, on-board aircraft connectivity is still limited. While the number of airlines offering in-flight connectivity increases, the current performance is insufficient to satisfy several hundreds of passengers simultaneously. There are several options to connect aircraft to the ground, i.e. direct air-to-ground, satellites and relaying via air-to-air links. However, each single solution is insufficient. The direct air-to-ground coverage is limited to the continent and coastal regions, while the satellite links are limited in the minimum size of the spot beams and air-to-air links need to be combined with a link to the ground. Moreover, even if a direct air-to-ground or satellite link is available, the peak throughput offered on each link is rarely achieved, as the capacity needs to be shared with other aircraft flying in the same coverage area. The main challenge in achieving a high throughput per aircraft lies in the throughput allocation. All aircraft should receive a fair share of the available throughput. More specifically, as an aircraft contains a network itself, a weighted share according to the aircraft size should be provided. To address this problem, an integrated air-to-ground network, which is able to provide a high throughput to aircraft, is proposed here. Therefore, this work introduces a weighted-fair throughput allocation scheme to provide such a desired allocation. While various aspects of aircraft connectivity are studied in literature, this work is the first to address an integrated air-to-ground network to provide high-throughput connectivity to aircraft. This work models the problem of throughput allocation as a mixed integer linear program. Two throughput allocation schemes are proposed, a centralized optimal solution and a distributed heuristic solution. For the optimal solution, two different objectives are introduced, a max-min-based and a threshold-based objective. The optimal solution is utilized as a benchmark for the achievable throughput for small scenarios, while the heuristic solution offers a distributed approach and can process scenarios with a higher number of aircraft. Additionally, an option for weighted-fair throughput allocation is included. Hence, large aircraft obtain a larger share of the throughput than smaller ones. This leads to fair throughput allocation with respect to the size of the aircraft. To analyze the performance of throughput allocation in the air-to-ground network, this work introduces an air-to-ground network model. It models the network realistically, but independent from specific network implementations, such as 5G or WiFi. It is also adaptable to different scenarios. The aircraft network is studied based on captured flight traces. Extensive and representative parameter studies are conducted, including, among others, different link setups, geographic scenarios, aircraft capabilities, link distances and link capacities. The results show that the throughput can be distributed optimally during high-aircraft-density times using the optimal solution and close to optimal using the heuristic solution. The mean throughput during these times in the optimal reference scenario with low Earth orbit satellites is 20 Mbps via direct air-to-ground links and 4 Mbps via satellite links, which corresponds to 10.7% and 1.9% of the maximum link throughput, respectively. Nevertheless, during low-aircraft-density times, which are less challenging, the throughput can reach more than 200 Mbps. Therefore, the challenge is on providing a high throughput during high-aircraft-density times. In the larger central European scenario, using the heuristic scheme, a minimum of 22.9 Mbps, i.e. 3.2% of the maximum capacity, can be provided to all aircraft during high-aircraft-density times. Moreover, the critical parameters to obtain a high throughput are presented. For instance, this work shows that multi-hop air-to-air links are dispensable for aircraft within direct air-to-ground coverage. While the computation time of the optimal solution limits the number of aircraft in the scenario, larger scenarios can be studied using the heuristic scheme. The results using the weighted-fair throughput allocation show that the introduction of weights enables a user-fair throughput allocation instead of an aircraft-fair throughput allocation. As a conclusion, using the air-to-ground model and the two introduced throughput allocation schemes, the achievable weighted-fair throughput per aircraft and the respective link choices can be quantified.