How many whole numbers lie in the interval between $\frac{5}{3}$ and $2\pi$?

$\textbf{(A)}\ 2\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ \text{infinitely many}$
1. **Determine the decimal values of the given fractions and constants:**
   \[
   \frac{5}{3} \approx 1.6667
   \]
   \[
   2\pi \approx 2 \times 3.14159 \approx 6.2832
   \]

2. **Identify the smallest whole number greater than \(\frac{5}{3}\):**
   \[
   \lceil 1.6667 \rceil = 2
   \]
   Here, \(\lceil x \rceil\) denotes the ceiling function, which rounds \(x\) up to the nearest whole number.

3. **Identify the largest whole number less than \(2\pi\):**
   \[
   \lfloor 6.2832 \rfloor = 6
   \]
   Here, \(\lfloor x \rfloor\) denotes the floor function, which rounds \(x\) down to the nearest whole number.

4. **List the whole numbers between the smallest and largest identified in steps 2 and 3:**
   \[
   2, 3, 4, 5, 6
   \]

5. **Count the whole numbers in the list:**
   \[
   2, 3, 4, 5, 6 \quad \text{(5 numbers)}
   \]

Conclusion:
\[
\boxed{5}
\]