**Core Concept:** Amniocentesis is a diagnostic procedure performed on pregnant women to obtain a sample of amniotic fluid for genetic testing, fetal cell analysis, or evaluation of other potential fetal abnormalities. In the context of this question, amniocentesis is performed to screen for Down's syndrome in a second pregnancy involving a mother with a child diagnosed with Down's syndrome.

**Why the Correct Answer is Right:** The appropriate time for amniocentesis to screen for Down's syndrome is during the **late second trimester**, typically between weeks 15-20 of gestation. This is when the risk of chromosomal abnormalities, including Down's syndrome, is highest. The procedure is generally safe and has a success rate of 95-98%, making it an effective screening tool for detecting genetic disorders.

**Why Each Wrong Option is Incorrect:** Option A (early first trimester) is incorrect because the risk of chromosomal abnormalities is relatively low during this period (weeks 5-14). Option B (early second trimester) is also incorrect as the risk is higher in the late second trimester. Option C (third trimester) is wrong because the risk of chromosomal abnormalities decreases significantly after week 28 of gestation, making it an unsuitable time for the procedure.

**Clinical Pearl:** Amniocentesis is generally safe and effective for screening Down's syndrome and other genetic disorders during the late second trimester (weeks 15-20). It is crucial to perform the procedure during this time window to maximize the chances of detecting potential abnormalities and to ensure the highest success rate of the test.