Sterling dropped to $1.2798 at one point in Asian trade, before recovering slightly to $1.2929.
The pound has now fallen by about 14 per cent against the dollar since hitting $1.50 ahead of the referendum result.
On the UK stock market, the FTSE 100 index opened down 11.47 points, or 0.2 per cent, at 6,533.90.
The FTSE 250 - which contains more UK-focused companies - fell 0.4 per cent to 15,671.47.
Earlier, Asian stock markets had closed lower, with Japan’s Nikkei index down 1.9 per cent.