  Coded Caching is an efficient technique to reduce peak hour network traffic.
One limitation of known coded caching schemes is that the demands of all users
are revealed to their peers in the delivery phase. Schemes that assure privacy
for user demands are studied in recent past. Assuming that the users are
equipped with caches of small memory sizes, the achievable rate under demand
privacy constraints is investigated in this work. We present an MDS code based
demand private coded caching scheme with $K$ users and $N$ files that achieves
a memory rate pair
$\left(\frac{1}{K(N-1)+1},N\left(1-\frac{1}{K(N-1)+1}\right)\right)$. The
presented memory-rate pair meets the lower bound under demand-privacy
requirements, proposed by Yan \textit{et al.} in the recent work \cite{c13}. By
memory sharing this characterizes the exact rate-memory trade-off for the
demand private coded caching scheme for cache memory $M\in
\left[0,\frac{1}{K(N-1)+1}\right]$.
