Voyager 1 has been in the reservoir of energetic ions and electrons that constitutes the heliosheath since it crossed the solar wind termination shock(1-3) on 16 December 2004 at a distance from the Sun of 94 astronomical units (1 AU = 1.5 x 10(8) km). It is now similar to 22 AU past the termination shock crossing(4). The bulk velocity of the plasma in the radial-transverse plane has been determined(5) using measurements of the anisotropy of the convected energetic ion distribution(6). Here we report that the radial component of the velocity has been decreasing almost linearly over the past three years, from similar to 70 km s(-1) to similar to 0 km s(-1), where it has remained for the past eight months. It now seems that Voyager 1 has entered a finite transition layer of zero-radial-velocity plasma flow, indicating that the spacecraft may be close to the heliopause, the border between the heliosheath and the interstellar plasma. The existence of a flow transition layer in the heliosheath contradicts current predictions(7)-generally assumed by conceptual models-of a sharp discontinuity at the heliopause.