diff --git a/Lib/zipfile.py b/Lib/zipfile.py
index 68051c8..dcebf72 100644
--- a/Lib/zipfile.py
+++ b/Lib/zipfile.py
@@ -346,7 +346,7 @@
         # compress_size         Size of the compressed file
         # file_size             Size of the uncompressed file
 
-    def FileHeader(self):
+    def FileHeader(self, zip64=None):
         """Return the per-file header as a string."""
         dt = self.date_time
         dosdate = (dt[0] - 1980) << 9 | dt[1] << 5 | dt[2]
@@ -362,12 +362,17 @@
         extra = self.extra
 
         min_version = 0
-        if file_size > ZIP64_LIMIT or compress_size > ZIP64_LIMIT:
-            # File is larger than what fits into a 4 byte integer,
-            # fall back to the ZIP64 extension
+        if zip64 is None:
+            zip64 = file_size > ZIP64_LIMIT or compress_size > ZIP64_LIMIT
+        if zip64:
             fmt = '<HHQQ'
             extra = extra + struct.pack(fmt,
                     1, struct.calcsize(fmt)-4, file_size, compress_size)
+        if file_size > ZIP64_LIMIT or compress_size > ZIP64_LIMIT:
+            if not zip64:
+                raise LargeZipFile("Filesize would require ZIP64 extensions")
+            # File is larger than what fits into a 4 byte integer,
+            # fall back to the ZIP64 extension
             file_size = 0xffffffff
             compress_size = 0xffffffff
             min_version = ZIP64_VERSION
@@ -1301,7 +1306,7 @@
             zinfo.CRC = 0
             self.filelist.append(zinfo)
             self.NameToInfo[zinfo.filename] = zinfo
-            self.fp.write(zinfo.FileHeader())
+            self.fp.write(zinfo.FileHeader(False))
             return
 
         cmpr = _get_compressor(zinfo.compress_type)
@@ -1309,8 +1314,11 @@
             # Must overwrite CRC and sizes with correct data later
             zinfo.CRC = CRC = 0
             zinfo.compress_size = compress_size = 0
-            zinfo.file_size = file_size = 0
-            self.fp.write(zinfo.FileHeader())
+            # Compressed size can be larger than uncompressed size
+            zip64 = self._allowZip64 and \
+                    zinfo.file_size * 1.05 > ZIP64_LIMIT
+            self.fp.write(zinfo.FileHeader(zip64))
+            file_size = 0
             while 1:
                 buf = fp.read(1024 * 8)
                 if not buf:
@@ -1330,11 +1338,16 @@
             zinfo.compress_size = file_size
         zinfo.CRC = CRC
         zinfo.file_size = file_size
-        # Seek backwards and write CRC and file sizes
+        if not zip64 and self._allowZip64:
+            if file_size > ZIP64_LIMIT:
+                raise RuntimeError('File size has increased during compressing')
+            if compress_size > ZIP64_LIMIT:
+                raise RuntimeError('Compressed size larger than uncompressed size')
+        # Seek backwards and write file header (which will now include
+        # correct CRC and file sizes)
         position = self.fp.tell()       # Preserve current position in file
-        self.fp.seek(zinfo.header_offset + 14, 0)
-        self.fp.write(struct.pack("<LLL", zinfo.CRC, zinfo.compress_size,
-              zinfo.file_size))
+        self.fp.seek(zinfo.header_offset, 0)
+        self.fp.write(zinfo.FileHeader(zip64))
         self.fp.seek(position, 0)
         self.filelist.append(zinfo)
         self.NameToInfo[zinfo.filename] = zinfo
@@ -1376,14 +1389,18 @@
             zinfo.compress_size = len(data)    # Compressed size
         else:
             zinfo.compress_size = zinfo.file_size
-        zinfo.header_offset = self.fp.tell()    # Start of header data
-        self.fp.write(zinfo.FileHeader())
+        zip64 = zinfo.file_size > ZIP64_LIMIT or \
+                zinfo.compress_size > ZIP64_LIMIT
+        if zip64 and not self._allowZip64:
+            raise LargeZipFile("Filesize would require ZIP64 extensions")
+        self.fp.write(zinfo.FileHeader(zip64))
         self.fp.write(data)
-        self.fp.flush()
         if zinfo.flag_bits & 0x08:
             # Write CRC and file sizes after the file data
-            self.fp.write(struct.pack("<LLL", zinfo.CRC, zinfo.compress_size,
+            fmt = '<LQQ' if zip64 else '<LLL'
+            self.fp.write(struct.pack(fmt, zinfo.CRC, zinfo.compress_size,
                   zinfo.file_size))
+        self.fp.flush()
         self.filelist.append(zinfo)
         self.NameToInfo[zinfo.filename] = zinfo
 
diff --git a/Misc/NEWS b/Misc/NEWS
index c444ed6..292138d 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -147,6 +147,9 @@
 Library
 -------
 
+- Issue #9720: zipfile now writes correct local headers for files larger than
+  4 GiB.
+
 - Issue #16955: Fix the poll() method for multiprocessing's socket
   connections on Windows.