Step 1: Understand the historical context.
This question is related to the chapter The Making of a Global World. During the sixteenth century, European powers such as Portugal and Spain expanded their trade and colonial activities across the world. They travelled to the Americas for trade, conquest and settlement. Along with goods, soldiers, sailors and missionaries, they also unknowingly carried diseases to the newly discovered lands.
Step 2: Identify the disease that spread in America.
The deadly disease that spread in America through European contact was smallpox. Europeans had already been exposed to smallpox in the Old World and many had developed some level of immunity. However, the native people of America had no previous exposure to this disease and therefore had almost no immunity against it.
Step 3: Explain the role of Portugal and Spain.
Portuguese and Spanish conquerors and traders reached the Americas during the age of exploration. Their arrival created direct contact between Europe and the American continent. This contact brought not only new goods and weapons but also germs and infections. Smallpox spread rapidly among the native population because they were biologically unprepared to resist it.
Step 4: Impact on Native Americans.
Smallpox proved disastrous for the indigenous population of America. It killed millions of Native Americans and destroyed many communities. The disease weakened local societies, reduced their population and made it easier for European powers to conquer and control American territories. Thus, disease became an invisible but powerful weapon of colonial expansion.
Step 5: Eliminate the incorrect options.
Yellow fever, malaria and cholera were also serious diseases in different historical contexts, but the disease specifically associated with European contact in America during the sixteenth century was smallpox. Therefore, Options (A), (C) and (D) are incorrect.
Step 6: Final conclusion.
Hence, the deadly disease that spread in America through Portugal and Spain in the mid-sixteenth century was:
\[
\boxed{\textbf{(B) Smallpox}}
\]