diff -r 62d844d2bd07 Lib/test/test_zipfile.py
--- a/Lib/test/test_zipfile.py	Fri May 13 21:19:22 2016 +0300
+++ b/Lib/test/test_zipfile.py	Fri May 13 22:10:58 2016 +0300
@@ -424,6 +424,49 @@ class StoredTestsWithSourceFile(Abstract
             f.seek(len(data))
             with zipfile.ZipFile(f, "r") as zipfp:
                 self.assertEqual(zipfp.namelist(), [TESTFN])
+                self.assertEqual(zipfp.read(TESTFN), self.data)
+        with open(TESTFN2, 'rb') as f:
+            self.assertEqual(f.read(len(data)), data)
+            zipfiledata = f.read()
+        with io.BytesIO(zipfiledata) as bio, zipfile.ZipFile(bio) as zipfp:
+            self.assertEqual(zipfp.namelist(), [TESTFN])
+            self.assertEqual(zipfp.read(TESTFN), self.data)
+
+    def test_read_concatenated_zip_file(self):
+        with io.BytesIO() as bio:
+            with zipfile.ZipFile(bio, 'w', zipfile.ZIP_STORED) as zipfp:
+                zipfp.write(TESTFN, TESTFN)
+            zipfiledata = bio.getvalue()
+        data = b'I am not a ZipFile!'*10
+        with open(TESTFN2, 'wb') as f:
+            f.write(data)
+            f.write(zipfiledata)
+
+        with zipfile.ZipFile(TESTFN2) as zipfp:
+            self.assertEqual(zipfp.namelist(), [TESTFN])
+            self.assertEqual(zipfp.read(TESTFN), self.data)
+
+    def test_append_to_concatenated_zip_file(self):
+        with io.BytesIO() as bio:
+            with zipfile.ZipFile(bio, 'w', zipfile.ZIP_STORED) as zipfp:
+                zipfp.write(TESTFN, TESTFN)
+            zipfiledata = bio.getvalue()
+        data = b'I am not a ZipFile!'*1000000
+        with open(TESTFN2, 'wb') as f:
+            f.write(data)
+            f.write(zipfiledata)
+
+        with zipfile.ZipFile(TESTFN2, 'a') as zipfp:
+            self.assertEqual(zipfp.namelist(), [TESTFN])
+            zipfp.writestr('strfile', self.data)
+
+        with open(TESTFN2, 'rb') as f:
+            self.assertEqual(f.read(len(data)), data)
+            zipfiledata = f.read()
+        with io.BytesIO(zipfiledata) as bio, zipfile.ZipFile(bio) as zipfp:
+            self.assertEqual(zipfp.namelist(), [TESTFN, 'strfile'])
+            self.assertEqual(zipfp.read(TESTFN), self.data)
+            self.assertEqual(zipfp.read('strfile'), self.data)
 
     def test_ignores_newline_at_end(self):
         with zipfile.ZipFile(TESTFN2, "w", zipfile.ZIP_STORED) as zipfp:
diff -r 62d844d2bd07 Lib/zipfile.py
--- a/Lib/zipfile.py	Fri May 13 21:19:22 2016 +0300
+++ b/Lib/zipfile.py	Fri May 13 22:10:58 2016 +0300
@@ -1140,10 +1140,10 @@ class ZipFile:
                 # even if no files are added to the archive
                 self._didModify = True
                 try:
-                    self.start_dir = self.fp.tell()
+                    self.start_dir = self._start_disk = self.fp.tell()
                 except (AttributeError, OSError):
                     self.fp = _Tellable(self.fp)
-                    self.start_dir = 0
+                    self.start_dir = self._start_disk = 0
                     self._seekable = False
                 else:
                     # Some file-like objects can provide tell() but not seek()
@@ -1164,7 +1164,7 @@ class ZipFile:
                     # set the modified flag so central directory gets written
                     # even if no files are added to the archive
                     self._didModify = True
-                    self.start_dir = self.fp.tell()
+                    self.start_dir = self._start_disk = self.fp.tell()
             else:
                 raise RuntimeError("Mode must be 'r', 'w', 'x', or 'a'")
         except:
@@ -1208,17 +1208,18 @@ class ZipFile:
         offset_cd = endrec[_ECD_OFFSET]         # offset of central directory
         self._comment = endrec[_ECD_COMMENT]    # archive comment
 
-        # "concat" is zero, unless zip was concatenated to another file
-        concat = endrec[_ECD_LOCATION] - size_cd - offset_cd
+        # self._start_disk:  Position of the start of ZIP archive
+        # It is zero, unless ZIP was concatenated to another file
+        self._start_disk = endrec[_ECD_LOCATION] - size_cd - offset_cd
         if endrec[_ECD_SIGNATURE] == stringEndArchive64:
             # If Zip64 extension structures are present, account for them
-            concat -= (sizeEndCentDir64 + sizeEndCentDir64Locator)
+            self._start_disk -= (sizeEndCentDir64 + sizeEndCentDir64Locator)
 
         if self.debug > 2:
-            inferred = concat + offset_cd
-            print("given, inferred, offset", offset_cd, inferred, concat)
+            inferred = self._start_disk + offset_cd
+            print("given, inferred, offset", offset_cd, inferred, self._start_disk)
         # self.start_dir:  Position of start of central directory
-        self.start_dir = offset_cd + concat
+        self.start_dir = offset_cd + self._start_disk
         fp.seek(self.start_dir, 0)
         data = fp.read(size_cd)
         fp = io.BytesIO(data)
@@ -1258,7 +1259,7 @@ class ZipFile:
                             t>>11, (t>>5)&0x3F, (t&0x1F) * 2 )
 
             x._decodeExtra()
-            x.header_offset = x.header_offset + concat
+            x.header_offset = x.header_offset + self._start_disk
             self.filelist.append(x)
             self.NameToInfo[x.filename] = x
 
@@ -1726,11 +1727,10 @@ class ZipFile:
                 file_size = zinfo.file_size
                 compress_size = zinfo.compress_size
 
-            if zinfo.header_offset > ZIP64_LIMIT:
-                extra.append(zinfo.header_offset)
+            header_offset = zinfo.header_offset - self._start_disk
+            if header_offset > ZIP64_LIMIT:
+                extra.append(header_offset)
                 header_offset = 0xffffffff
-            else:
-                header_offset = zinfo.header_offset
 
             extra_data = zinfo.extra
             min_version = 0
@@ -1777,7 +1777,7 @@ class ZipFile:
         # Write end-of-zip-archive record
         centDirCount = len(self.filelist)
         centDirSize = pos2 - self.start_dir
-        centDirOffset = self.start_dir
+        centDirOffset = self.start_dir - self._start_disk
         requires_zip64 = None
         if centDirCount > ZIP_FILECOUNT_LIMIT:
             requires_zip64 = "Files count"