diff --git a/Lib/zipfile.py b/Lib/zipfile.py
index ac17177..bb8fc53 100644
--- a/Lib/zipfile.py
+++ b/Lib/zipfile.py
@@ -163,6 +163,7 @@
         return endrec
 
     # Update the original endrec using data from the ZIP64 record
+    endrec[_ECD_SIGNATURE] = sig
     endrec[_ECD_DISK_NUMBER] = disk_num
     endrec[_ECD_DISK_START] = disk_dir
     endrec[_ECD_ENTRIES_THIS_DISK] = dircount
@@ -735,9 +736,8 @@
 
         # "concat" is zero, unless zip was concatenated to another file
         concat = endrec[_ECD_LOCATION] - size_cd - offset_cd
-        if endrec[_ECD_LOCATION] > ZIP64_LIMIT:
-            # If the offset of the "End of Central Dir" record requires Zip64
-            # extension structures, account for them
+        if endrec[_ECD_SIGNATURE] == stringEndArchive64:
+            # If Zip64 extension structures are present, account for them
             concat -= (sizeEndCentDir64 + sizeEndCentDir64Locator)
 
         if self.debug > 2: