diff --git a/dvc/external_repo.py b/dvc/external_repo.py
index a598eb6692..f5117332b6 100644
--- a/dvc/external_repo.py
+++ b/dvc/external_repo.py
@@ -248,7 +248,7 @@ def _clone_default_branch(url, rev):
         else:
             logger.debug("erepo: git clone %s to a temporary dir", url)
             clone_path = tempfile.mkdtemp("dvc-clone")
-            git = Git.clone(url, clone_path)
+            git = Git.clone(url, clone_path, rev=rev)
             CLONES[url] = clone_path
     finally:
         if git:
diff --git a/dvc/scm/git/__init__.py b/dvc/scm/git/__init__.py
index e70d6f7b0f..2c60f49207 100644
--- a/dvc/scm/git/__init__.py
+++ b/dvc/scm/git/__init__.py
@@ -1,5 +1,6 @@
 """Manages Git."""
 
+from functools import partial
 import logging
 import os
 import yaml
@@ -71,16 +72,31 @@ def clone(url, to_path, rev=None):
             # [1] https://github.com/gitpython-developers/GitPython/issues/924
             env[ld_key] = ""
 
+        clone_from = partial(
+            git.Repo.clone_from,
+            url,
+            to_path,
+            env=env,  # needed before we can fix it in __init__
+            no_single_branch=True,
+        )
+
         try:
-            tmp_repo = git.Repo.clone_from(
-                url,
-                to_path,
-                env=env,  # needed before we can fix it in __init__
-                no_single_branch=True,
-            )
-            tmp_repo.close()
+            if rev:
+                logger.debug("attempting shallow clone of branch %s", rev)
+                tmp_repo = clone_from(branch=rev, depth=1)
+            else:
+                logger.debug("full clone")
+                tmp_repo = clone_from()
         except git.exc.GitCommandError as exc:
-            raise CloneError(url, to_path) from exc
+            if not rev or ("Remote branch %s not found" % rev) not in str(exc):
+                raise CloneError(url, to_path) from exc
+            try:
+                logger.debug("not a branch - performing full clone")
+                tmp_repo = clone_from()
+            except git.exc.GitCommandError as exc:
+                raise CloneError(url, to_path) from exc
+
+        tmp_repo.close()
 
         # NOTE: using our wrapper to make sure that env is fixed in __init__
         repo = Git(to_path)