The West Indies moved up two spots above South Africa (seventh) and Sri Lanka (eighth), whom the Windies beat in last month’s series in the Caribbean, on the ICC rankings list.
South Africa's most recent Test cricket was a 2-0 away series defeat to fifth-ranked Pakistan in February.
India (1) and New Zealand (2) are still the top-ranked ICC Test nations.
England have moved above Australia into third position and Pakistan and West Indies are respectively in fifth and sixth posiion on the rankings list.
Behind South Africa are Sri Lanka (8), Bangladesh and Zimbabwe