(1) Background: loss of quality of life (QoL) and functional independence are two of the most common consequences of suffering a stroke. The main objective of this research is to study which factors are the greatest determinants of functional capacity and QoL a month after suffering a stroke so that they can be considered in early interventions. (2) Methods: a cross-sectional study was conducted which sample consisted of 81 people who had previously suffered a stroke. The study population was recruited at the time of discharge from the Neurology Service and Stroke Unit of the hospitals of Burgos and Córdoba, Spain, through a consecutive sampling. Data were collected one month after participants experienced a stroke, and the main study variables were quality of life, measured with the Stroke-Specific Quality of Life Measure (NEWSQOL), and functional independence, measured with the Functional Independence Measure-Functional Assessment Measure (FIM-FAM). (3) Results: the factors associated with a worse QoL and functional capacity one month after having suffered a stroke were living in a different dwelling than the usual flat or house (p < 0.05), a worse cognitive capacity (p < 0.001) and a worse functional capacity of the affected upper limb (p < 0.001). A higher age was related to a worse functional capacity one month after suffering a stroke (p = 0.048). (4) Conclusions: the type of dwelling, age, cognitive ability and functional capacity of the affected upper limb are determining aspects in functional independence and QoL during the first weeks after a stroke.