Step 1: Understanding the Concept:
The question tests an exception to the general rule that an agreement made without consideration is void. Specifically, it deals with a promise to pay a debt that has become legally unenforceable due to the expiry of the limitation period (a 'time-barred' debt).
Step 2: Detailed Explanation:
The general rule is laid down in Section 25 of the Indian Contract Act, 1872: "An agreement made without consideration is void". However, the same section provides certain exceptions.
\begin{itemize}
\item Section 25(3) creates a specific exception for time-barred debts. It states that a promise to pay a debt barred by the law of limitation is a valid and enforceable contract if the promise is:
\begin{enumerate}
\item made in writing, and
\item signed by the person to be charged (the debtor) or his authorized agent.
\end{enumerate}
\end{itemize}
In the given scenario:
\begin{itemize}
\item There is a time-barred debt.
\item Murli (the debtor) makes a promise to pay a part of it (Rs. 4,500).
\item The promise is in writing.
\item Murli signs the document.
\end{itemize}
All the conditions of Section 25(3) are fulfilled. Therefore, the new promise to pay Rs. 4,500 constitutes a valid and enforceable contract, even though there is no new consideration for it.
Step 3: Final Answer:
This contract is Enforceable.