Given the true statement: The picnic on Sunday will not be held only if the weather is not fair.  We can then conclude that:

$ \textbf{(A)}\ \text{If the picnic is held, Sunday's weather is undoubtedly fair.}$
$ \textbf{(B)}\ \text{If the picnic is not held, Sunday's weather is possibly unfair.}$
$ \textbf{(C)}\ \text{If it is not fair Sunday, the picnic will not be held.}$
$ \textbf{(D)}\ \text{If it is fair Sunday, the picnic may be held.}$
$ \textbf{(E)}\ \text{If it is fair Sunday, the picnic must be held.}$
Given the statement: "The picnic on Sunday will not be held only if the weather is not fair."

We can rephrase this statement in logical terms. Let:
- \( P \) represent "The picnic on Sunday will be held."
- \( Q \) represent "The weather is fair on Sunday."

The given statement can be written as:
\[ \neg P \rightarrow \neg Q \]

This means "If the picnic is not held, then the weather is not fair."

To find the correct conclusion, we need to consider the contrapositive of the given statement. The contrapositive of \( \neg P \rightarrow \neg Q \) is:
\[ Q \rightarrow P \]

This means "If the weather is fair, then the picnic will be held."

Now, let's analyze each option:

1. **(A) If the picnic is held, Sunday's weather is undoubtedly fair.**
   - This statement can be written as \( P \rightarrow Q \).
   - This is not necessarily true based on the given statement. The given statement does not imply \( P \rightarrow Q \).

2. **(B) If the picnic is not held, Sunday's weather is possibly unfair.**
   - This statement can be written as \( \neg P \rightarrow \neg Q \).
   - This is the given statement itself, but it says "possibly unfair," which is not a strong conclusion.

3. **(C) If it is not fair Sunday, the picnic will not be held.**
   - This statement can be written as \( \neg Q \rightarrow \neg P \).
   - This is the contrapositive of the given statement, which is logically equivalent to the given statement.

4. **(D) If it is fair Sunday, the picnic may be held.**
   - This statement can be written as \( Q \rightarrow P \) but with "may" instead of "must."
   - This is not a strong conclusion based on the given statement.

5. **(E) If it is fair Sunday, the picnic must be held.**
   - This statement can be written as \( Q \rightarrow P \).
   - This is the contrapositive of the given statement, which is logically equivalent to the given statement.

Therefore, the correct conclusion is:
\[ \boxed{E} \]