This paper presents an electroencephalographic (EEG) P300-based brain-computer interface (BCI) Internet browser. The system uses the "odd-ball" row-col paradigm for generating the P300 evoked potentials on the scalp of the user, which are immediately processed and translated into web browser commands. There were previous approaches for controlling a BCI web browser. However, to the best of our knowledge, none of them was focused on an assistive context, failing to test their applications with a suitable number of end users. In addition, all of them were synchronous applications, where it was necessary to introduce a "read-mode" command in order to avoid a continuous command selection. Thus, the aim of this study is twofold: 1) to test our web browser with a population of multiple sclerosis (MS) patients in order to assess the usefulness of our proposal to meet their daily communication needs; and 2) to overcome the aforementioned limitation by adding a threshold that discerns between control and non-control states, allowing the user to calmly read the web page without undesirable selections. The browser was tested with sixteen MS patients and five healthy volunteers. Both quantitative and qualitative metrics were obtained. MS participants reached an average accuracy of 84.14%, whereas 95.75% was achieved by control subjects. Results show that MS patients can successfully control the BCI web browser, improving their personal autonomy.