Mission San Diego de Alcalá was the first Mission built in 1769; and, Mission Carmel came second in 1770… presumably aligned with the founding of the Monterey Presidio and discovery of the San Francisco Bay in that same year.
The original Mission was close to the Presidio but almost immediately, Junípero Serra petitioned the Spanish viceroy and received permission to move the Mission to Carmel to “…shield their neophytes from soldiers and civilians, who exposed them to alcohol, gambling, venereal diseases, and notions of private property.” (A World Transformed: Firsthand Accounts of California Before the Gold Rush)
A new Mission was built at its current location (1770) and is today considered one of the most authentic. The bell tower dome is supposedly original and inside the church is the grave of Junípero Serra, near the front pulpit.
When we visited there was a fair bit of construction on the bell tower dome and some of the surrounding buildings. As such, we could not access the graveyard because it was littered with construction materials.