diff --git a/Lib/gzip.py b/Lib/gzip.py
index 983e0cee07f..08f9da5a5a9 100644
--- a/Lib/gzip.py
+++ b/Lib/gzip.py
@@ -5,7 +5,7 @@
 
 # based on Andrew Kuchling's minigzip.py distributed with the zlib module
 
-import struct, sys, time
+import struct, sys, time, os
 import zlib
 import builtins
 
@@ -158,7 +158,8 @@ def _write_gzip_header(self):
         try:
             # RFC 1952 requires the FNAME field to be Latin-1. Do not
             # include filenames that cannot be represented that way.
-            fname = self.name.encode('latin-1')
+            fname = os.path.basename(self.name)
+            fname = fname.encode('latin-1')
             if fname.endswith(b'.gz'):
                 fname = fname[:-3]
         except UnicodeEncodeError:
diff --git a/Misc/NEWS b/Misc/NEWS
index 030430a9fd8..a53b28cfc5e 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -120,6 +120,9 @@ C-API
 Library
 -------
 
+- Issue #4750: Store the basename of the original filename in the gzip FNAME
+  header as required by RFC 1952.
+
 - Issue #1180: Added a new global option to ignore ~/.pydistutils.cfg in
   Distutils.