The figures, based on the latest 12 weeks' sales through to April 22, confirm that Asda's share of trade is 15.4% against Sainsbury's 15.1%.
Neither supermarket came close to overtaking food giants Tesco, which continues to outperform the market with share of trade up to 29.1%, and growth of 5% on the previous year.
Asda traditionally does well over holiday periods such as Christmas and Easter, whereas Sainsbury's has a strong food offering, pitting them as strong contenders against each other.
Sainsbury's only edged ahead of rival Asda to become the UK's second largest supermarket at the end of March, before slipping back again, according to ACNielsen.
The ACNielsen findings are likely to be confirmed by data being released by Taylor Nelson Sofres next week. It too is understood to have Asda in the number two slot and likely to remain so, according to analysts who see Asda as being stronger during the summer months.
Analysts are predicting that it will not be until winter now that Sainsbury's will take the number two slot.
The findings come as Asda plans an overhaul of its advertising, in which England and Manchester United footballer Wayne Rooney and his girlfriend Coleen McLoughlin will become the new faces of the company in a deal worth £3.5m.
Eleni Nicholas, group managing director for marketing at ACNielsen UK and Ireland, said: "There is becoming somewhat of a photo-finish each month between Asda and Sainsbury's, as each of them maximise their unique strengths.
"All the supermarkets have made significant space for summer merchandise and it will be interesting to monitor the impact of this on share of trade over the summer months."
Morrisons takes fourth place with 10.7% of the share.
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