The group, which also operates Grosvenor Casinos, said revenues in the six months to the end of December were hit further by the hot weather in July.
Pre-tax profits fell 23% to £27.7 million after earnings plunged 38% at Mecca and 26% at its Grosvenor chain.
Chief executive Ian Burke said: "The very challenging bingo market has contributed to a decline in the Mecca brand's performance."
The group estimated the slump in Mecca admissions alone translated to a £4.6 million decline in Rank operating profits, the bulk of a £5.5 million decline overall.
Revenue at bingo halls dropped by 7% on a like-for-like basis, with customer visits down from 6.86 million to 6.21 million.
The 10% fall was "due to the general pressure on our customers' discretionary expenditures and the impact of the hot July weather", the group said.
Digital customer visits were down by 5%, and revenues by 2% though within this, mobile revenues surged by 24%.
The group said it was trying to stabilise and rebuild the performance of its bingo halls through improving value and restricting spending to essential works, as well as continuing to lobby for a cut in bingo duty.
Meanwhile, Grosvenor Casinos saw overall revenues increase by 34% after the acquisition of 19 venues from Gala in May.
However, like-for-like turnover was down 9%, after a weak London performance and the hot summer.
The group said operating profit in the second half of the year was expected to be flat on a year earlier, after trading in the first four weeks of January was in line with expectations and showed an improvement over the first half.
Rank said the performance of its businesses in the "challenging" recent period had been expected.
But analysts at Panmure Gordon said it was "a disappointing set of results".