diff --git a/Lib/test/test_zipfile64.py b/Lib/test/test_zipfile64.py
index a6f3dca..a87baaa 100644
--- a/Lib/test/test_zipfile64.py
+++ b/Lib/test/test_zipfile64.py
@@ -99,7 +99,7 @@
     def testMoreThan64kFiles(self):
         # This test checks that more than 64k files can be added to an archive,
         # and that the resulting archive can be read properly by ZipFile
-        zipf = zipfile.ZipFile(TESTFN, mode="w")
+        zipf = zipfile.ZipFile(TESTFN, mode="w", allowZip64=True)
         zipf.debug = 100
         numfiles = (1 << 16) * 3/2
         for i in xrange(numfiles):
@@ -111,8 +111,43 @@
         self.assertEqual(len(zipf2.namelist()), numfiles)
         for i in xrange(numfiles):
             self.assertEqual(zipf2.read("foo%08d" % i), "%d" % (i**3 % 57))
+        zipf2.close()
+
+    def testMoreThan64kFilesAppend(self):
+        zipf = zipfile.ZipFile(TESTFN, mode="w", allowZip64=False)
+        zipf.debug = 100
+        numfiles = (1 << 16) - 1
+        for i in range(numfiles):
+            zipf.writestr("foo%08d" % i, "%d" % (i**3 % 57))
+        self.assertEqual(len(zipf.namelist()), numfiles)
+        with self.assertRaises(zipfile.LargeZipFile):
+            zipf.writestr("foo%08d" % numfiles, b'')
+        self.assertEqual(len(zipf.namelist()), numfiles)
         zipf.close()
 
+        zipf = zipfile.ZipFile(TESTFN, mode="a", allowZip64=False)
+        zipf.debug = 100
+        self.assertEqual(len(zipf.namelist()), numfiles)
+        with self.assertRaises(zipfile.LargeZipFile):
+            zipf.writestr("foo%08d" % numfiles, b'')
+        self.assertEqual(len(zipf.namelist()), numfiles)
+        zipf.close()
+
+        zipf = zipfile.ZipFile(TESTFN, mode="a", allowZip64=True)
+        zipf.debug = 100
+        self.assertEqual(len(zipf.namelist()), numfiles)
+        numfiles2 = (1 << 16) * 3//2
+        for i in range(numfiles, numfiles2):
+            zipf.writestr("foo%08d" % i, "%d" % (i**3 % 57))
+        self.assertEqual(len(zipf.namelist()), numfiles2)
+        zipf.close()
+
+        zipf2 = zipfile.ZipFile(TESTFN, mode="r")
+        self.assertEqual(len(zipf2.namelist()), numfiles2)
+        for i in range(numfiles2):
+            self.assertEqual(zipf2.read("foo%08d" % i), "%d" % (i**3 % 57))
+        zipf2.close()
+
     def tearDown(self):
         test_support.unlink(TESTFN)
         test_support.unlink(TESTFN2)
diff --git a/Lib/zipfile.py b/Lib/zipfile.py
index fe71765..6722c40 100644
--- a/Lib/zipfile.py
+++ b/Lib/zipfile.py
@@ -30,7 +30,7 @@
 error = BadZipfile      # The exception raised by this module
 
 ZIP64_LIMIT = (1 << 31) - 1
-ZIP_FILECOUNT_LIMIT = 1 << 16
+ZIP_FILECOUNT_LIMIT = (1 << 16) - 1
 ZIP_MAX_COMMENT = (1 << 16) - 1
 
 # constants for Zip file compression methods
@@ -1101,12 +1101,17 @@
         if zinfo.compress_type not in (ZIP_STORED, ZIP_DEFLATED):
             raise RuntimeError, \
                   "That compression method is not supported"
-        if zinfo.file_size > ZIP64_LIMIT:
-            if not self._allowZip64:
-                raise LargeZipFile("Filesize would require ZIP64 extensions")
-        if zinfo.header_offset > ZIP64_LIMIT:
-            if not self._allowZip64:
-                raise LargeZipFile("Zipfile size would require ZIP64 extensions")
+        if not self._allowZip64:
+            requires_zip64 = None
+            if len(self.filelist) >= ZIP_FILECOUNT_LIMIT:
+                requires_zip64 = "Files count"
+            elif zinfo.file_size > ZIP64_LIMIT:
+                requires_zip64 = "Filesize"
+            elif zinfo.header_offset > ZIP64_LIMIT:
+                requires_zip64 = "Zipfile size"
+            if requires_zip64:
+                raise LargeZipFile(requires_zip64 +
+                                   " would require ZIP64 extensions")
 
     def write(self, filename, arcname=None, compress_type=None):
         """Put the bytes from filename into the archive under the name
@@ -1256,10 +1261,8 @@
 
         try:
             if self.mode in ("w", "a") and self._didModify: # write ending records
-                count = 0
                 pos1 = self.fp.tell()
                 for zinfo in self.filelist:         # write central directory
-                    count = count + 1
                     dt = zinfo.date_time
                     dosdate = (dt[0] - 1980) << 9 | dt[1] << 5 | dt[2]
                     dostime = dt[3] << 11 | dt[4] << 5 | (dt[5] // 2)
@@ -1320,13 +1323,21 @@
 
                 pos2 = self.fp.tell()
                 # Write end-of-zip-archive record
-                centDirCount = count
+                centDirCount = len(self.filelist)
                 centDirSize = pos2 - pos1
                 centDirOffset = pos1
-                if (centDirCount >= ZIP_FILECOUNT_LIMIT or
-                    centDirOffset > ZIP64_LIMIT or
-                    centDirSize > ZIP64_LIMIT):
+                requires_zip64 = None
+                if centDirCount > ZIP_FILECOUNT_LIMIT:
+                    requires_zip64 = "Files count"
+                elif centDirOffset > ZIP64_LIMIT:
+                    requires_zip64 = "Central directory offset"
+                elif centDirSize > ZIP64_LIMIT:
+                    requires_zip64 = "Central directory size"
+                if requires_zip64:
                     # Need to write the ZIP64 end-of-archive records
+                    if not self._allowZip64:
+                        raise LargeZipFile(requires_zip64 +
+                                           " would require ZIP64 extensions")
                     zip64endrec = struct.pack(
                             structEndArchive64, stringEndArchive64,
                             44, 45, 45, 0, 0, centDirCount, centDirCount,
diff --git a/Misc/NEWS b/Misc/NEWS
index 39f6fce..5b69a1b 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -22,6 +22,9 @@
 Library
 -------
 
+- Issue #21866: ZipFile.close() no longer writes ZIP64 central directory
+  records if allowZip64 is false.
+
 - Issue #22415: Fixed debugging output of the GROUPREF_EXISTS opcode in the re
   module.