Stroke remains a leading cause of death and disability. A limited number of therapies, such as intravenous tissue plasminogen activator, have been approved to interrupt stroke in the early hours after symptom onset. Many patients are not able to benefit from these therapies, however, and so a need exists for development of new interventions to reduce disability after stroke. This study will be an early step towards this, and will examine the safety of two cell types, mononuclear cells and marrow stromal cells. In each case, the cells will be autologous, specifically being derived from the subject's own bone marrow.