A new modality for treatment of cancer involves the ex vivo growth of cancer-specific T-cells for subsequent infusion into the patient. The therapeutic aim is selective destruction of cancer cells by the activated infused cells. An important problem in the early phase of developing such a treatment is to determine a maximal tolerated dose (MTD) for use in a subsequent phase II clinical trial. Dose may be quantified by the number of cells infused per unit body weight, and determination of an MTD may be based on the probability of infusional toxicity as a function of dose. As in a phase I trial of a new chemotherapeutic agent, this may be done by treating successive cohorts of patients at different dose levels, with each new level chosen adaptively based on the toxicity data of the patients previously treated. Such a dose-finding strategy is inadequate in T-cell infusion trials because the number of cells grown ex vivo for a given patient may be insufficient for infusing the patient at the current targeted dose. To address this problem, we propose an algorithm for trial conduct that determines a feasible MTD based on the probabilities of both infusibility and toxicity as functions of dose. The method is illustrated by application to a dendritic cell activated lymphocyte infusion trial in the treatment of acute leukemia. A simulation study indicates that the proposed methodology is both safe and reliable.