diff --git a/Lib/tarfile.py b/Lib/tarfile.py
index e2b6053..6769066 100755
--- a/Lib/tarfile.py
+++ b/Lib/tarfile.py
@@ -1249,6 +1249,8 @@
 
             length, keyword = match.groups()
             length = int(length)
+            if length == 0:
+                raise InvalidHeaderError("invalid header")
             value = buf[match.end(2) + 1:match.start(1) + length - 1]
 
             # Normally, we could just use "utf-8" as the encoding and "strict"
diff --git a/Lib/test/recursion.tar b/Lib/test/recursion.tar
new file mode 100644
index 0000000..b823725
--- /dev/null
+++ b/Lib/test/recursion.tar
Binary files differ
diff --git a/Lib/test/test_tarfile.py b/Lib/test/test_tarfile.py
index d60d35b..3ddeb97 100644
--- a/Lib/test/test_tarfile.py
+++ b/Lib/test/test_tarfile.py
@@ -429,6 +429,13 @@
                 with self.assertRaisesRegex(tarfile.ReadError, "unexpected end of data"):
                     tar.extractfile(t).read()
 
+    def test_length_zero_header(self):
+        # bpo-39017 (CVE-2019-20907): reading a zero-length header should fail
+        # with an exception
+        with self.assertRaisesRegex(tarfile.ReadError, "file could not be opened successfully"):
+            with tarfile.open(support.findfile('recursion.tar')) as tar:
+                pass
+
 class MiscReadTestBase(CommonReadTest):
     def requires_name_attribute(self):
         pass