diff --git a/archive/tar/reader.go b/archive/tar/reader.go
index 0b0c3b1..dbc5698 100644
--- a/archive/tar/reader.go
+++ b/archive/tar/reader.go
@@ -899,6 +899,9 @@ func (sfr *sparseFileReader) Read(b []byte) (n int, err error) {
 		// Otherwise, we're at the end of the file
 		return 0, io.EOF
 	}
+	if sfr.tot < sfr.sp[0].offset {
+		return 0, io.ErrUnexpectedEOF
+	}
 	if sfr.pos < sfr.sp[0].offset {
 		// We're in a hole
 		n = sfr.readHole(b, sfr.sp[0].offset)
diff --git a/archive/tar/reader_test.go b/archive/tar/reader_test.go
index ab1e844..6ffb383 100644
--- a/archive/tar/reader_test.go
+++ b/archive/tar/reader_test.go
@@ -757,3 +757,22 @@ func TestNegativeHdrSize(t *testing.T) {
 	}
 	io.Copy(ioutil.Discard, r)
 }
+
+// This used to hang in (*sparseFileReader).readHole due to missing
+// verification of sparse offsets against file size.
+func TestIssue10968(t *testing.T) {
+	f, err := os.Open("testdata/issue10968.tar")
+	if err != nil {
+		t.Fatal(err)
+	}
+	defer f.Close()
+	r := NewReader(f)
+	_, err = r.Next()
+	if err != nil {
+		t.Fatal(err)
+	}
+	_, err = io.Copy(ioutil.Discard, r)
+	if err != io.ErrUnexpectedEOF {
+		t.Fatalf("expected %q, got %q", io.ErrUnexpectedEOF, err)
+	}
+}
diff --git a/archive/tar/testdata/issue10968.tar b/archive/tar/testdata/issue10968.tar
new file mode 100644
index 0000000..1cc837b
Binary files /dev/null and b/archive/tar/testdata/issue10968.tar differ