The anatomical structure that secures the tooth within the alveolar bone socket is the periodontal ligament.

The periodontal ligament is a specialized connective tissue that forms a fibrous attachment between the cementum covering the tooth root and the alveolar bone. Its primary function is to anchor the tooth firmly in the socket, while also allowing for slight movement, which helps in absorbing forces during chewing. This ligament contains collagen fibers called Sharpey's fibers that insert into both the cementum and alveolar bone, providing a strong, flexible connection.

In contrast:
- The gingival sulcus is the space or crevice between the free gingiva and the tooth surface; it does not provide any supportive or securing function.
- The alveolar mucosa is the movable mucous membrane lining the alveolar bone but has no role in tooth attachment.
- The cementoenamel junction is the anatomical boundary between the enamel covering the crown and the cementum covering the root of the tooth; it serves as a landmark but does not secure the tooth into the bone.

Therefore, the periodontal ligament is essential for tooth support and attachment within the alveolar socket.

Reference: Gray's Anatomy, 42nd Edition, Head and Neck, Chapter 16: Oral Cavity and Teeth / Page 850-855