Step 1: Recall the limitation of an ordinary diode rectifier.
A silicon diode requires approximately
\[
0.7\,\text{V}
\]
before it conducts.
Hence, very small input signals cannot be rectified accurately.
Step 2: State the advantage of a precision rectifier.
A precision rectifier uses an op-amp to eliminate the effect of the diode's forward voltage drop.
Thus, it can accurately rectify very small signals.
Therefore,
\[
\boxed{\text{Accurate rectification of small signals}}
\]
is the correct answer.
\[
\boxed{(A)}
\]