Final answer:
The security of Florida and the southernmost areas of Southeast America in 1819 was achieved through the Adams-Onís Treaty, in which Spain ceded Florida to the United States.
Step-by-step explanation:
In 1819, Florida and the southernmost areas of Southeast America were secured by the United States through the Adams-Onís Treaty. This treaty was instrumental in the expansion of U.S. territories, whereby Spain ceded Florida to the United States in exchange for the U.S. settling claims over Texas and the payment of $5 million to Spain. Furthermore, the treaty delineated the boundary between New Spain and the United States, which also led to America renouncing its claims to Texas, at least temporarily.
The desire to acquire Florida was driven by various factors, including the sheltering of runaway slaves and the strategic need to quell conflicts with native tribes like the Creek and Seminole Nations. Andrew Jackson's military excursions into Spanish territory and the inability of Spain to control the region effectively under the pressure of independence movements in Latin America were instrumental in pushing both nations towards the negotiation table, leading to the signing of the treaty.