Round off 1,627.083 to the nearest one decimal place.

The number is 1,627.083. The digit in the first decimal place (tenths place) is 0.

The digit immediately to the right of 0 is 8.

If the digit to the right is 5 or greater, round up the digit in the first decimal place. If it is less than 5, keep the digit as it is.
Since 8 is greater than or equal to 5, we round up the 0.

Rounding up 0 makes it 1. So, 1,627.083 rounded to the nearest one decimal place is 1,627.1.

Comparing this with the given options:
A. 1,627.0
B. 1,627.1
C. 1,627.8
D. 1,627.08

The correct option is B.
The answer is $\boxed{\text{B. } 1,627.1}$.