The rupee resumed barely weak at 63.78 as against its previous closing of 63.75 at the Interbank Foreign Exchange (Forex) market.
It moved in a range of 63.72 and 63.78 during the morning deals before quoting at 63.76 at 1030 hours.
The US dollar index was down 0.09 per cent to 97.81 in early trade.
In New York, the dollar closed the session just below a two-month high against the yen yesterday after a gauge of service sector growth from the Institute for Supply Management rose to its highest level in 10 years.
Meanwhile, the benchmark 30-share index was trading up 30 points, or 0.11 per cent, at 28,253.56 at 1035 hours.