This study is an open-label, multiple dose, dose-escalation protocol to evaluate the safety and preliminary efficacy of an orally administered drug, DMP777, which reduces the inflammation in the lungs caused by excessive activity of white blood cells called neutrophils. The excessive inflammatory activity is due to a chemical called elastase; DMP777 blocks this activity, and is thus an anti-inflammatory drug. In this protocol, patients with cystic fibrosis are evaluated for the presence of the drug in lung fluids by obtaining samples of lung fluids using a technique called bronchoalveolar lavage, these fluids are also assessed for levels of inflammatory cells and chemicals at baseline and after receiving the drug. The body's handling of metabolism of DMP777 is assessed by measures of lung and blood levels after multiple doses.