16. A computer program product for handling message threads, the computer program product comprising: one or more computer-readable tangible storage devices; program instructions, stored on at least one of the one or more storage devices, to display a primary dialog window, wherein the primary dialog window includes a first ongoing conversation pane and a first outgoing message pane; program instructions, stored on at least one of the one or more storage devices, to display, in the primary dialog window, a first discussion between a first party and a second party during a chat session between the first party and the second party; program instructions, stored on at least one of the one or more storage devices, to, during the chat session, receive an indication, from the first party, of a new thread, wherein the new thread is a subset of the chat session, and wherein the new thread is a second discussion between the first party and the second party; program instructions, stored on at least one of the one or more storage devices, to, after receiving the indication, display a secondary dialog window, wherein the secondary dialog window includes a second ongoing conversation pane and a second outgoing message pane; program instructions, stored on at least one of the one or more storage devices, to display responses to first outgoing messages sent from the first outgoing message pane in the first ongoing conversation pane; and program instructions, stored on at least one of the one or more storage devices, to display responses to second outgoing messages sent from the second outgoing message pane in the second ongoing conversation pane.
19. The computer program product according to claim 16 further comprising: program instructions, stored on at least one of the one or more storage devices, to, after receiving the indication, send a message to an instant messaging program on a remote computer being used by the second party to open a window for the new thread.