Given an array of numbers $A = (672, 673, 674, ..., 2016)$ on table. 
Three arbitrary numbers $a,b,c \in A$ are step by step replaced by number $\frac13 min(a,b,c)$. 
After $672$ times, on the table there is only one number $m$, such that

(A): $0 < m < 1$ (B): $m = 1$ (C): $1 < m < 2$ (D): $m = 2$ (E): None of the above.
1. **Initial Setup**: We start with the array \( A = (672, 673, 674, \ldots, 2016) \). The array contains consecutive integers from 672 to 2016. The total number of elements in \( A \) is \( 2016 - 672 + 1 = 1345 \).

2. **Replacement Process**: In each step, we replace three arbitrary numbers \( a, b, c \) with \( \frac{1}{3} \min(a, b, c) \). This process reduces the number of elements in the array by 2 each time (since we start with 3 numbers and replace them with 1).

3. **Number of Steps**: We need to determine how many steps are required to reduce the array to a single number. Since each step reduces the number of elements by 2, we need \( 1344 \) steps to reduce \( 1345 \) elements to 1 element:
   \[
   1345 - 2 \times 672 = 1
   \]
   Therefore, \( 672 \) steps are required.

4. **Behavior of the Replacement**: In each step, the smallest of the three chosen numbers is divided by 3. This means that the smallest number in the array will be repeatedly divided by 3.

5. **Final Number Calculation**: The smallest number in the initial array is 672. After \( 672 \) steps, the smallest number will be divided by \( 3^{672} \):
   \[
   m = 672 \times \left( \frac{1}{3} \right)^{672}
   \]
   Since \( \left( \frac{1}{3} \right)^{672} \) is an extremely small number, \( m \) will be a very small positive number.

6. **Range of \( m \)**: Given that \( m \) is a very small positive number, it will definitely be less than 1. Therefore, the correct answer is:
   \[
   0 < m < 1
   \]

The final answer is (A): \( \boxed{ 0 < m < 1 } \)