If they were to be defeated in next month's re-match, the Hammers would miss out on the chance of millions of pounds in revenue that could be generated by winning at Wembley and taking a place in the Uefa Cup.
Although West Ham have insisted that fielding the ineligible striker, Manny Omoyinmi, was a "genuine administration error", Mackrell has shouldered the blame. Having only arrived at Upton Park last summer from Sheffield Wednesday to replace Peter Storrie, he will leave the club as soon as a replacement has been found. The club's football secretary, Alison O'Dowd, has also resigned.
The club's chairman, Terence Brown, said: "We feel Graham has made an honourable decision but it seems a high price to pay for what was a small and genuine administration error."
Mackrell described his decision as "sad but somewhat inevitable" once the Football League had ordered a re-play after Omoyinmi, who came on as a late substitute during last Wednesday's penalty shoot-out victory over Villa, was later found to have been cup-tied. The striker had played in both legs of an earlier round while on loan at Gillingham, but apparently he did not think to mention that fact to anyone at the club, according to West Ham's manager, Harry Redknapp.
Mackrell continues to insist that West Ham had made inquiries at Gillingham after the end of the loan period and were told that the striker had not been cup-tied.
However, officials of the Second Division club are understood not to agree completely with West Ham's version of events, although they made no official statement yesterday.
Mackrell declared: "I felt in the light of the circumstances regarding the decision to make the club replay that I am the person who is ultimately responsible for administration and it was the honourable thing for me to offer my resignation to the board. I'm conscious of the supporters' disappointment at the game having to be replayed and in the light of that, I felt that something along these lines was appropriate.
"Eventually you have to take some responsibility in life for the actions that go on and the people that work for you.
"Obviously we would not have played the player unless we'd thought he was eligible to play. It has transpired that information was wrong and we are now paying a heavy price for that. It was a little error but unfortunately it has had large consequences for the club."
Asked whether Omoyinmi should share the blame, he told West Ham ClubCall: "You would have to ask him that. At the end of the day, he's paid to play football and that's all water under the bridge now."
The re-match has been scheduled to take place on Tuesday 11 January, with those supporters from both clubs who still have their ticket-stubs from the original encounter able to obtain replacement tickets for the replay for free.
Other spectators are likely to have to pay the full admission price, although West Ham are expected to donate to charity any profits realised after they have covered the cost of stewarding and policing.
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