You can use a temporary table

CREATE TABLE #DeletedRows(
    [ID] [INT] NOT NULL,
    [Code] [VARCHAR](16) NULL,
    [Title] [NVARCHAR](128) NULL
)

DELETE SourceTable 
OUTPUT 
    DELETED.[ID], 
    DELETED.[Code],
    DELETED.[Title]
INTO #DeletedRows([ID], [Code], [Title])
WHERE Condition

INSERT ArchiveTable([OldID], [Code], [Title])
OUTPUT INSERTED.*
SELECT [ID], [Code], [Title]
FROM #DeletedRows

DROP TABLE #DeletedRows

A variant with a table variable

DECLARE @DeletedRows TABLE(
    [ID] [INT] NOT NULL,
    [Code] [VARCHAR](16) NULL,
    [Title] [NVARCHAR](128) NULL
)

DELETE SourceTable 
OUTPUT 
    DELETED.[ID], 
    DELETED.[Code],
    DELETED.[Title]
INTO @DeletedRows([ID], [Code], [Title])
WHERE Condition

INSERT ArchiveTable([OldID], [Code], [Title])
OUTPUT INSERTED.*
SELECT [ID], [Code], [Title]
FROM @DeletedRows

I found an interesting variant using DML with OUTPUT in SP and INSERT...EXEC... after that:

Test tables:

CREATE TABLE TestTable(
  ID int NOT NULL PRIMARY KEY,
  Title varchar(10) NOT NULL
)

CREATE TABLE TestTableLog(
  LogID int NOT NULL IDENTITY,
  OperType char(1) NOT NULL,
  CHECK(OperType IN('I','U','D')),
  ID int NOT NULL,
  Title varchar(10) NOT NULL
)

DML procedures:

CREATE PROC InsTestTable
  @ID int,
  @Title varchar(10)
AS

  INSERT TestTable(ID,Title)
  OUTPUT inserted.ID,inserted.Title,'I' OperType
  VALUES(@ID,@Title)

GO

CREATE PROC UpdTestTable
  @ID int,
  @Title varchar(10)
AS

  UPDATE TestTable
  SET
    Title=@Title
  OUTPUT inserted.ID,inserted.Title,'U' OperType
  WHERE ID=@ID

GO

CREATE PROC DelTestTable
  @ID int
AS

  DELETE TestTable
  OUTPUT deleted.ID,deleted.Title,'D' OperType
  WHERE ID=@ID

GO

Tests:

-- insert test
INSERT TestTableLog(ID,Title,OperType)
EXEC InsTestTable 1,'A'

INSERT TestTableLog(ID,Title,OperType)
EXEC InsTestTable 2,'B'

INSERT TestTableLog(ID,Title,OperType)
EXEC InsTestTable 3,'C'

-- update test
INSERT TestTableLog(ID,Title,OperType)
EXEC UpdTestTable 2,'BBB'

-- delete test
INSERT TestTableLog(ID,Title,OperType)
EXEC DelTestTable 3
GO

-- show resutls
SELECT *
FROM TestTableLog

Maybe it'll be interesting to someone.

I encountered the same issue and this resolved it. First make sure you set this on the target table;

SET IDENTITY_INSERT [schema].[target_table_name] ON

And then define all the columns in both the DELETED section and the INTO list;

DELETE [schema].[source_table]
 OUTPUT DELETED.[col1],
        DELETED.[col2],
        DELETED.[col3] 
    INTO [dest_table] ([col1],
                       [col2],
                       [col3])  
        FROM [schema].[source_table]