Real included Denis Cheryshev in their starting lineup in a last-32, first-leg game at third-tier Cadiz in early December when he should have been serving a one-match ban. The Russia forward opened the scoring in a 3-1 victory.
The TAD, which had already turned down attempts by Real Madrid to get their expulsion temporarily suspended, said in a written ruling it had rejected the appeal and that the decision was final.
Read said they respected but disagreed with the decision, adding in a statement that they would now seek to appeal through the courts and were studying other legal options.
Real president Florentino Perez had previously said they did not break the rules and that neither Cheryshev, who incurred the sanction playing on loan at Villarreal last season, nor the club were notified that he was banned.
As well as the disqualifying Real from the King's Cup, the Spanish football federation (RFEF) fined them 6,001 euros (£4334), saying they should have been aware of the ban.
(Reporting by Sarah White, editing by Ed Osmond)
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