We simulate the formation of a BN fullerene from an amorphous B cluster at 2000 K by quantum mechanical molecular dynamics based on the density-functional tight-binding method. We run 30 trajectories 200 ps in length, where N atoms are supplied around the target cluster, which is initially an amorphous B36 cluster. Most of the incident N atoms are promptly incorporated into the target cluster to form B-N-B bridges or NB3 pyramidal local substructures. BN fullerene formation is initiated by alternating BN ring condensation. Spontaneous atomic rearrangement and N2 dissociation lead to the construction of an sp(2) single-shelled structure, during which the BN cluster undergoes a transition from a liquid-like to a solid-like state. Continual atomic rearrangement and sporadic N2 dissociation decrease the number of defective rings in the BN cluster and increase the number of six-membered rings, forming a more regular shell structure. The number of four-membered rings tends to remain constant, and contributes to more ordered isolated-tetragon-rule ring placement.