An apparatus for splicing a leading end of a first sliver deposited in a coiler can and a trailing end of an out-running second sliver fed into a fiber processing machine, includes a device for tapering the leading and trailing ends to obtain gradually tapered length portions thereof; a device for positioning the tapered length portions side by side to obtain a juxtapositioned relationship thereof; a pressure-applying constriction through which the first and second slivers pass for splicing together the leading and trailing ends by pressure; and a device for introducing the first sliver, spliced to the second sliver, into the fiber processing machine.