This thesis is a feasibility study for a crewed mission to a Near Earth Asteroid (NEA). An alternate dual-launch architecture is proposed and analyzed against a more established architecture. Instead of a rendezvous in a low-Earth parking orbit, the new architecture performs the rendezvous while the two spacecraft are on an Earth-escape trajectory to the destination NEA. After selecting a target asteroid, 2000 SG344, each architecture will have its best mission compared to the best mission of the other architecture. Using the new architecture, a mission is created to the chosen NEA, 2000 SG344. A back-up Orion MPCV and a Habitation Module are launched first on a cargo configuration SLS. A crew of two astronauts is launched two hours later in the primary Orion MPCV by a crewed configuration SLS. Both of these launches are on an Earth-escape trajectory and begin rendezvous after two full days in outer space. The completed spacecraft journeys the rest of the trip to the NEA. For a period of eight days, the spacecraft remains in a tight control sphere near the asteroid by using a control algorithm and the rendezvous thrusters. The astronauts have this period to perform their EVAs and accomplish their mission objectives at the NEA. The spacecraft then departs the NEA and returns to Earth. The entire mission is 134 days and requires 2.054 km/s of Delta-v maneuvers to complete. An analysis of multiple Lambert's methods is also done due to their extensive use in this thesis. Many of the most popular Lambert algorithms are compared by evaluating each on its accuracy, speed, and singularities. The best Lambert method to use for the orbital analysis in this paper is Battin's method because it is accurate, quick, and robust for all cases that will be observed.