Using this undocumented option suggested by [MichaelE2](http://mathematica.stackexchange.com/a/25949/9490), we can change the color of the border to the same as the background color of the plot:

    Manipulate[
     SphericalPlot3D[1, {theta, 0, Pi}, {phi, t, 2 Pi + t},
          Mesh -> {7, 7},
          MaxRecursion -> ControlActive[2, 5],
          PlotRange -> All,
          Boxed -> False,
          Axes -> None,
          SphericalRegion -> True,
          Method -> {"RotationControl" -> "Globe"},
          ImageSize -> {400, 400},
          Background -> Black],
     {{t, 0, Style["t", 10]}, 0, 2 Pi, 0.01}, 
     Method -> {"ContentAreaBackground" -> Black}]

[![enter image description here][1]][1]

But this option does not work for version 7, so it is then necessary to set the `FrameMargins` to a small negative value,

    Manipulate[
       SphericalPlot3D[1, {theta, 0, Pi}, {phi, t, 2 Pi + t},
            Mesh -> {7, 7},
            MaxRecursion -> ControlActive[2, 5],
            PlotRange -> All,
            Boxed -> False,
            Axes -> None,
            SphericalRegion -> True,
            Method -> {"RotationControl" -> "Globe"},
            ImageSize -> {400, 400},
            Background -> Black],
       {{t, 0, Style["t", 10]}, 0, 2 Pi, 0.01},
       FrameMargins -> #
       ] & /@ {Automatic, None, -5}

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/4M2oJ.png
  [2]: https://i.sstatic.net/DqAuD.png