diff --git a/CHANGELOG.md b/CHANGELOG.md
index a359d8f..ec41c10 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2,7 +2,8 @@
 
 # Master
 
-- Output the installed version of pip and setuptools in the build log (#1007).
+- Install an explicit version of wheel rather than the latest release at the time (#1007).
+- Output the installed version of pip, setuptools and wheel in the build log (#1007).
 - Install setuptools from PyPI rather than a vendored copy (#1007).
 - Reduce the number of environment variables exposed to `bin/{pre,post}_compile` and other subprocesses (#1011)
 
diff --git a/bin/steps/python b/bin/steps/python
index d297b37..4fca67b 100755
--- a/bin/steps/python
+++ b/bin/steps/python
@@ -138,10 +138,12 @@ fi
 
 PIP_VERSION='20.0.2'
 SETUPTOOLS_VERSION='39.0.1'
+WHEEL_VERSION='0.34.2'
 
 if [[ "${PYTHON_VERSION}" == ${PY34}* ]]; then
-  # Python 3.4 support was dropped in pip 19.2+.
+  # Python 3.4 support was dropped in pip 19.2+ and wheel 0.34.0+.
   PIP_VERSION='19.1.1'
+  WHEEL_VERSION='0.33.6'
 fi
 
 if [[ -f "$BUILD_DIR/Pipfile" ]]; then
@@ -163,13 +165,13 @@ fi
 # If a new Python has been installed or Pip isn't up to date:
 if [ "$FRESH_PYTHON" ] || [[ ! $(pip --version) == *${PIP_VERSION}* ]]; then
 
-  puts-step "Installing pip ${PIP_VERSION} and setuptools ${SETUPTOOLS_VERSION}"
+  puts-step "Installing pip ${PIP_VERSION}, setuptools ${SETUPTOOLS_VERSION} and wheel ${WHEEL_VERSION}"
 
   # Remove old installations.
   rm -fr /app/.heroku/python/lib/python*/site-packages/pip-*
   rm -fr /app/.heroku/python/lib/python*/site-packages/setuptools-*
 
-  /app/.heroku/python/bin/python "$GETPIP_PY" pip=="${PIP_VERSION}" "setuptools==${SETUPTOOLS_VERSION}" &> /dev/null
+  /app/.heroku/python/bin/python "$GETPIP_PY" pip=="${PIP_VERSION}" "setuptools==${SETUPTOOLS_VERSION}" "wheel==${WHEEL_VERSION}" &> /dev/null
 fi
 
 set -e
diff --git a/test/run-features b/test/run-features
index 1c059c8..478ada1 100755
--- a/test/run-features
+++ b/test/run-features
@@ -44,7 +44,7 @@ testStandardRequirements() {
 
 testPipenv() {
   compile "pipenv"
-  assertCaptured "Installing pip 9.0.2 and setuptools 39.0.1"
+  assertCaptured "Installing pip 9.0.2, setuptools 39.0.1 and wheel 0.34.2"
   assertCapturedSuccess
 }
 
diff --git a/test/run-versions b/test/run-versions
index 49775d8..09587d8 100755
--- a/test/run-versions
+++ b/test/run-versions
@@ -9,7 +9,7 @@ testPythonDefault() {
   compile "pythonDefault"
   assertCaptured $DEFAULT_PYTHON_VERSION
   assertNotCaptured "security update"
-  assertCaptured "Installing pip 20.0.2 and setuptools 39.0.1"
+  assertCaptured "Installing pip 20.0.2, setuptools 39.0.1 and wheel 0.34.2"
   assertCaptured "Installing SQLite3"
   assertCapturedSuccess
 }
@@ -25,7 +25,7 @@ testPython2() {
       assertNotCaptured "python-2-7-eol-faq";
     fi
     assertNotCaptured "security update"
-    assertCaptured "Installing pip 20.0.2 and setuptools 39.0.1"
+    assertCaptured "Installing pip 20.0.2, setuptools 39.0.1 and wheel 0.34.2"
     assertCaptured "Installing SQLite3"
     assertCapturedSuccess
 }
@@ -53,7 +53,7 @@ testPython3_4() {
   compile "python3_4"
   assertCaptured $LATEST_34
   assertNotCaptured "security update"
-  assertCaptured "Installing pip 19.1.1 and setuptools 39.0.1"
+  assertCaptured "Installing pip 19.1.1, setuptools 39.0.1 and wheel 0.33.6"
   # if cedar 14 and legacy binaries, fail. if cedar 14 and staging, succeed.
   if [[ ! -n $USE_STAGING_BINARIES ]] && [[ $STACK == "cedar-14" ]]; then
       assertCapturedError
@@ -89,7 +89,7 @@ testPython3_5() {
   compile "python3_5"
   assertCaptured $LATEST_35
   assertNotCaptured "security update"
-  assertCaptured "Installing pip 20.0.2 and setuptools 39.0.1"
+  assertCaptured "Installing pip 20.0.2, setuptools 39.0.1 and wheel 0.34.2"
   assertCaptured "Installing SQLite3"
   assertCapturedSuccess
 }
@@ -112,7 +112,7 @@ testPython3_6() {
   compile "python3_6"
   assertCaptured $LATEST_36
   assertNotCaptured "security update"
-  assertCaptured "Installing pip 20.0.2 and setuptools 39.0.1"
+  assertCaptured "Installing pip 20.0.2, setuptools 39.0.1 and wheel 0.34.2"
   assertCaptured "Installing SQLite3"
   assertCapturedSuccess
 }
@@ -139,7 +139,7 @@ testPython3_7() {
   else
     assertNotCaptured "security update"
     assertCaptured $LATEST_37
-    assertCaptured "Installing pip 20.0.2 and setuptools 39.0.1"
+    assertCaptured "Installing pip 20.0.2, setuptools 39.0.1 and wheel 0.34.2"
     assertCaptured "Installing SQLite3"
     assertCapturedSuccess
   fi
@@ -183,7 +183,7 @@ testPython3_8() {
   else
     assertNotCaptured "security update"
     assertCaptured $LATEST_38
-    assertCaptured "Installing pip 20.0.2 and setuptools 39.0.1"
+    assertCaptured "Installing pip 20.0.2, setuptools 39.0.1 and wheel 0.34.2"
     assertCaptured "Installing SQLite3"
     assertCapturedSuccess
   fi
@@ -202,7 +202,7 @@ testPypy3_6() {
   else
     assertCaptured "Installing pypy"
     assertCaptured "$PYPY_36"
-    assertCaptured "Installing pip 20.0.2 and setuptools 39.0.1"
+    assertCaptured "Installing pip 20.0.2, setuptools 39.0.1 and wheel 0.34.2"
     assertCapturedSuccess
   fi
 }
@@ -214,7 +214,7 @@ testPypy2_7() {
   else
     assertCaptured "Installing pypy"
     assertCaptured "$PYPY_27"
-    assertCaptured "Installing pip 20.0.2 and setuptools 39.0.1"
+    assertCaptured "Installing pip 20.0.2, setuptools 39.0.1 and wheel 0.34.2"
     assertCapturedSuccess
   fi
 }